algorithms and visualization for solutions of nonlinear ...j.zhou/cnz.pdf · algorithms and...

International Journal of Bifurcation and Chaos, Vol. 10, No. 7 (2000) 1565–1612c© World Scientific Publishing Company

ALGORITHMS AND VISUALIZATION FOR SOLUTIONSOF NONLINEAR ELLIPTIC EQUATIONS

GOONG CHEN∗ and JIANXIN ZHOU†

Department of Mathematics, Texas A&M University,College Station, TX 77843, USA

WEI-MING NI‡

School of Mathematics, University of Minnesota,Minneapolis, MN 55455, USA

Received August 18, 1999; Revised September 8, 1999

In this paper, we compute and visualize solutions of several major types of semilinear ellipticboundary value problems with a homogeneous Dirichlet boundary condition in 2D. We presentthe mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iter-ation and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are wellknown to be rich in their multiplicity of solutions. Many such physically significant solutionsare also known to lack stability and, thus, are elusive to capture numerically. We will computeand visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects ofthe domains on the multiplicity. Special emphasis is placed on SIA and MPA, by which multipleunstable solutions are computed. The domains include the disk, symmetric or nonsymmetricannuli, dumbbells, and dumbbells with cavities. The nonlinear partial differential equationsinclude the Lane–Emden equation, Chandrasekhar’s equation, Henon’s equation, a singularlyperturbed equation, and equations with sublinear growth. Relevant numerical data of solutionsare listed as possible benchmarks for other researchers. Commentaries from the existing litera-ture concerning solution behavior will be made, wherever appropriate. Some further theoreticalproperties of the solutions obtained from visualization will also be presented.

1. Introduction

In this paper, we study semilinear elliptic boundaryvalue problems (BVPs) of the form

∆u(x) + f(x, u(x)) = 0, x ∈ Ω,

u(x) = 0, x ∈ ∂Ω,(1)

where Ω is a bounded open domain in RN , N = 2,and f is a nonlinear function of x and u. We willdeal with f ≡ up, −u + up, or variants thereof.

We wish to compute numerical solutions of (1) andplot their graphics for visualization. In particular,we want to visualize multiple solutions of (1) on do-mains with various geometries and topologies. Wealso hope to survey existing algorithms and to intro-duce new ones, set certain numerical benchmarks,explore singular perturbation cases, and perhapseven “discover” theorems through visualization.

Models of (1) arise naturally in physics, engi-neering, biology and ecology, geometry, etc. Al-though nonlinearities may appear in seemingly

∗Supported in part by NSF Grant DMS 96-10076.E-mail: [email protected]†E-mail: [email protected]‡Supported in part by NSF Grant DMS 97-05639.

1565

1566 G. Chen et al.

endless forms, the simplest, yet most basic formof nonlinearity is the power type. In this con-nection, we may first mention the Lane–Emden(–Fowler) equation in astrophysics [Chandrasekhar,1939, Chap. 3] and [Fowler, 1931]

∆u+ up = 0 , u > 0 , on Ω, p > 1 , u|∂Ω = 0 ,(2)

where up is proportional to the density of thegaseous star [Chandrasekhar, 1939]. From the pointof view of analysis, this equation is interesting andchallenging because the Laplace operator ∆ is “neg-ative” while the nonlinear operator u 7→ up is “pos-itive” in appropriate function spaces, e.g. in H1

0(Ω),where

H10(Ω) = v ∈ H1(Ω)|v = 0 on ∂Ω

and Hs(Ω) denotes the Sobolev space W s,2(Ω) forgiven s ∈ R [Adams, 1975]. There is a “competi-tion” between these linear and nonlinear operators.In order for a solution to exist, a certain “balance” isrequired. There arises the prospect of either nonex-istence of solutions, or of the existence of possiblymultiple solutions, for the general semilinear equa-tion (1). Any nontrivial solution of (2) is an unsta-ble solution. (A solution of (1) is said to be unsta-ble if it is not a local maximum or minimum of afunctional corresponding to a canonical variationalformulation of (1); nor is it a steady state belongingto a corresponding time-dependent parabolic prob-lem whose governing equation is ut − ∆u + f = 0with the same boundary condition and certain ini-tial conditions.)

In contrast, if problem (1) is of the form∆u(x)−|u(x)|p−1u(x)=g(x), p>1,

g is given, on Ω,

u=0 on ∂Ω,(3)

i.e. the sign in front of the nonlinearity |u|p−1u in(2) is adjusted from “+” to “−”, then both the lin-ear and nonlinear operators on the left of the firstequation in (3) are “negative” and, as we can ex-pect, a rather different theory applies. Indeed, theexistence and uniqueness of solutions of (3) are wellestablished; see the monotone dissipative operatortheory in [Lions, 1969], for example. Some numer-ical solutions, along with graphics for those solu-tions of (3), may be found in [Deng et al., 1996,pp. 974–975]. Otherwise, we will not discuss muchabout (3) in this paper.

Let us return to (2) and to astrophysics; the do-main Ω with the most physical interest is BR, theopen ball of radius R in R3 centered at the origin.When Henon [1973] studied rotating stellar struc-tures, he proposed a variant of (2) as follows:

∆u+ |x|`up = 0 , u > 0 , on Ω ,

p > 1 ; ` > 0 ; u|∂Ω = 0 .(4)

(For his choice of nonlinearity, Henon commented:“This choice, although arbitrary, has the advan-tages of simplicity and convenience.”) Lieb and Yau[1987] considered Chandrasekhar’s theory of stel-lar collapse. They showed that the Chandrasekharequation for the white dwarf problem without thegeneral relativistic effect is equivalent to the follow-ing equation

∆u+ 4π(2u + u2)3/2 = 0 , in BR . (5)

Thus this equation is referred to as theChandrasekhar equation in this paper. In both[Chandrasekhar, 1939] and [Lieb & Yau, 1987], theprimary interest is in radial solutions u ≥ 0. Theseequations provide some of the physical backgroundfor the model (1). Certain equations in catalysistheory also can be described in the form (1), butthey often contain a varying parameter [Aris, 1975]and thus manifest bifurcation phenomena. Otherreaction–diffusion models such as Gierer and Mein-hardt’s system in biology (see [Geirer & Meinhardt,1972] and [Ni, 1998]), have a “strong relationship”,in a certain asymptotic sense to equations of theform (1).

The motivation for the model (1), along withits significance in other applications, will be givenin due course, below.

The mathematical theory of semilinear ellipticequations has attracted the interest of many ana-lysts and applied mathematicians. The subject areahas undergone an explosive growth since the Seven-ties and, consequently, there is a huge body of liter-ature. To review just some small specialized portionof such publications would require a gigantic effort;this will not be a task we either wish or can affordto undertake in this article. Rather, our focus of at-tention lies in algorithms for, and visualization of,solutions of (1) for certain types of nonlinearity asmentioned.

Concerning the algorithmic development for so-lutions of semilinear elliptic BVPs, we must firstcite the following two prominent theoretical meth-ods, which actually have formed the foundation formuch of the existing numerical study. These are:

Algorithms and Visualization for Solutions of Nonlinear Elliptic Equations 1567

(1) The Mountain Pass Lemma (MPL), publishedin [Ambrosetti & Rabinowitz, 1973]. It providesa minimax variational formulation of criticalpoints of functionals that are neither boundedfrom above nor below, such as those which cor-respond to (2). MPL is a powerful method forsemilinear elliptic BVPs, as well as for a host ofother nonlinear PDEs.

(2) The Monotone Iteration Scheme (MIS), alsocalled the barrier method or the method ofsuper- and sub-solutions. Its origin can betraced to Bieberbach, who used this method inmany of his papers; see [Bieberbach, 1916], forexample. It is a general, constructive methodfor finding stable solutions of semilinear ellipticequations [Sattinger, 1971]. Generalizations ofMIS to quasilinear elliptic BVPs, and to sys-tems of coupled elliptic semilinear PDEs calledquasimonotone iteration (QMI), are also avail-able. See [Amann, 1976; Amann & Crandall,1978; Chen et al., Part III, in review], and [Pao,1992], for example.

The mathematical proof of MPL as given in[Ambrosetti & Rabinowitz, 1973] contains bothconstructive and not-so-constructive elements and,therefore algorithmic realization of MPL is by nomeans straightforward. (In contrast, numerical im-plementation of MIS is much more direct and ob-vious.) However, Choi and McKenna [1993, 1996]were able to devise ingenious algorithms by fusingthe finite element method (FEM) with the methodof steepest descent, and obtain multiple solutions ofnonlinear elliptic PDEs on a rectangle. They andtheir collaborators have also successfully appliedMPL-based algorithms to suspension bridge andtraveling wave problems [Humphreys & McKenna,1999] and [Chen & McKenna, 1997] and to waterwaves [Hill, 1997]. In our opinion, a more appro-priate term for the algorithm devised by Choi andMcKenna is the “Mini–Max Algorithm” rather thanthe “Mountain–Pass Algorithm”. An explanationwill be given in Remark 2.2 in Sec. 2.

Numerical computations for solutions of semi-linear elliptic single equations or systems by MISor QMI can be implemented by iterations where, ateach iteration, a linear elliptic BVP is solved. Com-putationally, there is the choice of three basic typesof linear elliptic numerical solvers: FDM, FEM andBEM (the boundary element method). Numericalanalysis and results in this direction using FDM

may be found in [Huy et al., 1986; Pao, 1987, 1992,1995], etc. A special case using FEM may be foundin [Ishikara, 1984]. The case using BEM may befound in [Sakakihara, 1987; Deng et al., 1996; Chenet al., 1999].

We note that a semilinear elliptic BVP (1)will not be simultaneously solvable by both MPLand MIS, due to the discriminating nature of MPLfor unstable solutions, whereas MIS is mostly forstable solutions. It is also mainly due to this reasonthat proofs of convergence and rates can be givenfor MIS or QMI, but not for MPL. As we can seefrom Sec. 3.4 below, for example, it is possible thata problem (2) can have a continuous one-parameterfamily of uncountably many solutions, each of whichis infinitely close to neighboring solutions. There isno point in trying to give a convergence proof forsuch a case, since one never knows which solutionthe iterates are converging to. The best hope onecan shoot for is to establish convergence for isolatedcritical points by assuming that such critical pointsare nondegenerate; see Remark 3.1. We will makemore comments on numerical analysis and conver-gence in the survey below.

Obviously, MPL and MIS and their numericalimplementation are not the only feasible methodsfor computing solutions of (1). We may mentionNewton’s method , which is a generic method forsolving nonlinear problems, and can be applied invariational or other settings in order to treat (1).Actually, Newton’s is already a major numericalmethod for solving nonlinear ODEs. One may read[Keller, 1968] to find out how Newton’s method isincorporated into the shooting method to approx-imate solutions of two-point BVPs of ODEs. Inthe case of nonlinear elliptic PDEs, the applica-tion of Newton’s method to the corresponding vari-ational functional appears quite straightforward atfirst. However, a basic assumption to use Newton’smethod to find u satisfying J ′(u) = 0, i.e. a crit-ical point u of a functional J in a Banach space,is that J ′′(u) be invertible (implying nondegener-acy) as a bounded linear operator. But degeneracyis very common when multiple critical points areencountered. Thus, Newton’s method is ruled out,at least, for all those degenerate critical points inSec. 3.4 below. On the other hand, when a localminimization technique is used in a quasi-Newton’smethod, the search will lead to a solution with alocally minimized variational value, which is zeroin the case of (1). From our subsequent discus-sions in Examples 2.1 and 2.2 in Sec. 2, it actually

1568 G. Chen et al.

becomes apparent that without using local struc-ture of a critical point, e.g. imposing constraintssuch as (7) and (13) based on the theoretical argu-ment, Newton’s method alone is not likely to suc-ceed. Besides, “good choices” of the initial iteratemay not always be available because, in general, wedo not know beforehand whether a solution exists,let alone its possible profile.

Now, let us further address the numerical anal-ysis aspect of elliptic BVPs. If the elliptic equationor system is linear, then the BVP has an inherentcoercive structure known as the Lax–Milgram The-orem and the Garding inequality, that quickly leadsto the existence and uniqueness (or the contrary, ifcompatibility conditions are violated) of solutions.If the BVPs are discretized in a satisfactory wayby FDM, FEM or BEM, then the coercive struc-ture is inherited by the discretized problems, whichcan then be used to establish convergence as wellas to derive error estimates. For nonlinear ellip-tic problems, when they are of the monotone dissi-pative type such as (3), this coercive structure re-mains intact. Otherwise, in order to take advan-tage of the coercive structure, one has to assumethat nonlinearities are “mild”, i.e. that they sat-isfy a certain global Lipschitz condition so that theycan be absorbed by coercive terms, in much thesame way as perturbations. To quote from [Choi &McKenna, 1993]: “. . . A typical restriction was that∂f(x, u)/∂u < λ1, the first eigenvalue of the (nega-tive) Laplacian with Dirichlet boundary conditions.This type of assumption guaranteed existence anduniqueness of the solution and allowed the proof. . . . None of these approaches gave much insightinto how to numerically find solutions of boundaryvalue problems when there were multiple solutionsof a nonobvious type. . . .”. Their commentary,forthright yet incisive, is also quite agreeable to us.

Other than those [Chen et al., to appear;Deng et al., 1996; Huy, 1986; Ishihara, 1984; Pao,1987, 1992, 1995; Sakakihara, 1987] previously men-tioned, there is also a body of important existing lit-erature on the numerical analysis of semilinear andquasilinear elliptic BVPs concentrating on error es-timates with respect to Sobolev and Holder spacenorms. A few of those papers, containing importantcontributions, are cited below:

(1) [Bers, 1958; Parter, 1965; Greenspan & Parter,1965], for FDM;

(2) [Ciarlet et al., 1967; Dupont & Douglas, 1975;Brenner & Scott, 1994], for FEM.

At present, even though the results therein are notdirectly applicable to the problems considered herelike (2), because of the difficulties described above,the error estimates and algorithms given by thoseauthors above and elsewhere (such as [Eydeland &Spruck, 1988], for example) are quite delicate, andmany ideas remain useful. We believe they willeventually help the analysis of convergence and er-rors for problems of MPL types in this paper.

Historically, numerical analysis and computa-tional methods were developed with a major aimto aid in the investigation of physical phenomena.At the heart of contemporary computer aided re-search is visualization. This is especially true forthe study undertaken here, because visualizationgreatly enhances one’s intuition and helps organizeone’s thinking, particularly for nonlinear phenom-ena, where the investigation of pattern formation isthe chief objective. Scientists and applied mathe-maticians routinely have the need to compute andvisualize solutions for various types of nonlinearPDEs; they have published their work in many jour-nals, in vastly different disciplines. However, forsemilinear elliptic BVPs (1), although theoreticalresults abound, very few documented, concrete nu-merical results and graphics have been publishedbeyond those in the pioneering work of [Choi &McKenna, 1993]. Systematic, organized efforts tovisualize solutions of (1) thus seem to be largelylacking, to the best of our knowledge.

In order to compute and visualize solutions ofpartial differential equations, a significant amountof work is involved in the pre- and post-processingof domain geometry and solution data, algorithmicand coding development, testing, debugging and re-fining. This work requires long-term commitmentand large, concerted manpower. Consider FEM[Brenner & Scott, 1994; Ciarlet, 1978; Strang &Fix, 1973], for example. The processing of domaindata and grid-generation (triangulation) is simplestif the geometry is rectangular. Also, by buildingthereupon, one can nicely treat piecewise rectangu-lar (or triangular) domains such as those which areL-shaped, T -shaped or dumbbell-shaped, or rect-angular domains with triangular cavities, or com-binations thereof. From the point of view of themathematical study of nonlinear elliptic BVPs, suchdomains are not interesting to the theorists, forthe obvious reason that the boundaries have zerocurvature. (The curvature effect becomes particu-larly poignant in the computation and visualizationof nonlinear Neumann BVPs; see a sequel [Chen


et al., Part II, in preparation].) However, if thedomain has a curvilinear boundary, then the cod-ing work for the FEM triangulation and mesh re-finement increases substantially. The adjustmentof geometries is definitely not a trivial task, as faras FEM is concerned. In comparison, BEM worksmuch better in this regard in 2D.

In the study of numerical solutions of ellipticequations, the authors’ favorite choice of method isBEM. One of the major advantages of BEM is thatit is highly adaptable to the change of geometries,especially for two-dimensional (2D) problems. Twoof the authors, Chen and Zhou, began their visu-alization study of solutions of eigenvalue problemsof the Laplacian ∆ and bi-Laplacian ∆2 in [Chen& Zhou, 1993a, 1993b; Chen et al., 1994]. Subse-quently, the authors collaborated on numerical algo-rithms and visualization for nonlinear PDEs [Denget al., 1996; Chen et al., 1999]. One of the ma-jor algorithms of the paper, called the scaling itera-tive algorithm (SIA) in Sec. 2.2, has been found tobe particularly effective for semilinear elliptic BVPslike (2) where the nonlinearity is of the power type.We first began its development in 1993. At thattime, we were not aware of the mountain–pass algo-rithm (MPA) developed two years earlier by [Choi& McKenna, 1993], which was published in 1993 butcame to our attention in 1995. For a large numberof examples computed here, these two algorithms,MPA and SIA, produce equally accurate numericalsolutions (see Example 2.1 in Sec. 2). They caneven work together. They also serve as a corrobo-ration for each other. These two algorithms formthe backbone of this paper.

The major objectives of visualization here areto “see” the profiles’ locations of concentration andthe multiplicity of solutions of nonlinear ellipticBVPs subject to the change of geometry and topol-ogy of the domain. We have chosen domains belowin (59) in such a way that they have varying degreesof nonsymmetry, nonconvexity, non-starshapednessand multiconnectedness. (The only intended ex-ception is the domain Ω1 in (59)(i), the unit disk,which is symmetric, convex, starshaped and sim-ply connected. It is chosen mainly for the pur-pose of setting numerical benchmarks.) We havealso tried to take into account some asymptotic or“CAD/CAM” concepts by considering the possibil-ity of pushing some geometry or parameter to anextreme, such as the pathological annular domainΩ4 in (64), the singular perturbation case treatedin Sec. 4, or the case of large exponent (i.e. power)

p of the Lane–Emden equation in Sec. 5.3. Toreduce somewhat the overburdening coding work,all the domains Ωi, i = 1, 2, . . . , 9, in Secs. 3–5are chosen to have piecewise circular and/or linearboundaries. Although the boundary curves couldhave been chosen to contain more complex piece-wise quadratic segments, we feel that such work isnot necessary since our chosen domains already in-clude quite enough geometrical and topological fea-tures for the visualization of many basic effects. Wehave made BEM our method of choice for its adapt-ability to changes of geometry, as mentioned. Onlypiecewise constant boundary elements (i.e. panels)will be used for its simplicity and thus the “error-aversion” feature in computer coding. Even thoughwe have not adopted more recent methods such asmultigrids in our paper and in the computer pro-grams, we feel that our work serves as a beginning,a motivation for other researchers to explore usefulnew directions, such as multigrids (see [Bramble,1993], for example) for nonlinear PDEs.

The organization of our paper is as follows. InSec. 2, we introduce numerical algorithms, meth-ods, error estimators and some theoretical foun-dation. In Sec. 3, we compute and illustrate so-lutions of the Lane–Emden equation with powerp = 3, for various geometries, to great length. InSec. 4, we study a singularly perturbed equationand display the spike-layer pattern of solutions. InSec. 5, we compute the equations of Henon, Chan-drasekhar and Lane–Emden (with “large” powers)for selected geometries. Finally, in Sec. 6, westudy the equation with sublinear growth, basedon DIA, along with certain monotonicity propertiesof solutions observed by us. Commentaries, when-ever available and appropriate, are given alongsidethe graphics to aid in our understanding of thesenonlinear PDEs.

By nature of the way the paper is written andthat the graphical results are presented, we need tomake a review/survey of a large body of literature(which is mostly theoretical work on nonlinear el-liptic PDEs), even though we feel that this paperis primarily original research rather than a review.In the bibliography, we hope that we have includedat least those articles that are most directly rele-vant to our study and interests here; we apologizein advance for any inadvertent omissions. We alsoreiterate that BEM, the numerical method for solv-ing linear elliptic PDEs employed here, is not thefocal point of the paper. Other major linear ellipticsolvers–FEM and FDM–should work equally well

1570 G. Chen et al.

by any computational science researchers who pre-fer FEM or FDM over BEM.

2. Iterative Algorithms andNumerical Methods

In principle, any constructive proof should be real-izable into a useful computational scheme for nu-merical solutions. Therefore, the true issue is: Howefficient is that computational scheme in compari-son with other viable ones? If the coding work istoo involved, or if the requirement of CPU mem-ory and time is beyond capacity, then that schemebecomes unattractive or even unworthwhile. Nowa-days, the CPU time in a high performance worksta-tion is virtually costless. The choice of an algorith-mic development and the actual programming task,thus, in our opinion, depends mostly on the levelof ease or difficulty of coding by the programmerand analyst . The following two examples providesome condensed arguments, alternative to MPL, asto why the solutions of semilinear elliptic PDEs inSecs. 3–5 exist . The arguments therein are all con-structive. However, we will explain why such con-structive proofs are not suitable or compatible withour numerical approach (based on BEM) here and,therefore, are not converted into algorithms by usfor practical computational purposes.

Example 2.1. A constrained minimizationmethod for the Lane–Emden equation (2).

The following argument is now standard; see[Ni, 1987, pp. 18–20]. We include it here for thebenefit of those who are not nonlinear PDE spe-cialists. Let p satisfy 1 < p < (N + 2)/(N − 2),where N is the space dimension of Ω. Consider theminimization problem

infv∈C

∫Ω|∇v|2dx , (6)

where C, the constraint set, is defined to be

C =

v ∈ H1

0 (Ω)|∫

Ω|v|p+1dx = 1

. (7)

C is well-defined by the Sobolev Imbedding Theo-rem. Let vk be a minimizing sequence for (6):∫

Ω|∇vk|2dx→ λ ≡ inf

v∈C

∫Ω|∇v|2dx ,

∫Ω|vk|p+1dx = 1 , k = 1, 2 .

Since (∫

Ω |∇v|2dx)1/2 defines an equivalent H1-norm in H1

0(Ω), the sequence vk is boundedin H1

0(Ω). Therefore it contains a subsequencevk weakly converging to some u in H1

0(Ω). Bythe compact imbedding of H1

0(Ω) in Lp+1(Ω), wehave strong convergence in Lp+1(Ω) since p + 1 <2N/(N − 2). Thus

∫Ω |u|p+1dx = 1. On the other

hand, since vk → u weakly in H10(Ω),∫

Ω|∇u|2dx ≤ lim

k→∞

∫Ω|∇vk|2dx ≤ λ . (8)

Therefore u is a minimizer for (6) in C. We claimthat u ≥ 0, since otherwise we simply replace u by|u|. Also, ∇u 6≡ 0, and thus λ > 0, since u = 0 on∂Ω and

∫Ω |u|p+1dx = 1.

Now, from standard arguments in calculus ofvariations one easily concludes that u is a weaksolution of

∆u+ αup = 0 on Ω, u|∂Ω = 0 , (9)

where α is the Lagrange multiplier. From ellip-tic regularity estimates [Gilbarg & Trudinger, 1983]one sees that u is a classical solution of (9). We con-clude that u > 0 by the usual maximum principle[Protter & Weinberger, 1967]. Further, multiplying(9)1 by u and integrating by parts, we obtain

λ =

∫Ω|∇u|2dx = α

∫Ω|u|p+1dx = α .

Setting u = λ1/(p−1)u, we have solved (2).

Example 2.2. Let f ∈ C1(R) such that

f ′(t) >f(t)

t, ∀ t > 0 . (10)

Then [Ding & Ni, 1989] shows that a solution of themore general problem

∆u+ f(u) = 0 , u > 0 on Ω, u|∂Ω = 0 , (11)

exists, which is a solution of the following con-strained minimization problem

infv∈M

1

2

∫Ω

[|∇v|2 − F (v)]dx , (12)

where

M≡v∈H1

0(Ω)|v 6≡0,

∫Ω

[|∇v|2−vf(v)]dx=0

,

F (t)=

∫ t

0f(s)ds ;

(13)


M is the solution manifold, an idea first due to[Nehari, 1960].

Let us attempt to find numerical solutions of (2)or (11) by directly using the arguments suggestedin Examples 2.1 and 2.2. Because of the variationalformulations given there, FEM becomes the naturalmethod to be used. A family of finite dimensionalapproximating spaces Vn = spanφhi |1 ≤ i ≤ n(h),0 < h ≤ h0, of subspaces of H1

0 (Ω) is then chosensuch that Vh → H1

0(Ω) as the mesh size h decreasesto zero. In proceeding to do the minimization prob-lem (6) or (12), one must first choose functions

vh =

n(h)∑j=1

cjφhj , vh ∈ Vh ,

to be admissible in C (resp. M):∫Ω

∣∣∣∑ cjφhj

∣∣∣p+1

dx=1 ,

(resp.

∫Ω

[∣∣∣∇∑ cjφhj

∣∣∣2−∑ cjφhj f(∑

cjφhj

)]2

dx=0

).

(14)

Resolving the constraint relations in (14)explicitly requires triangulation of the entire do-main and extensive quadratures. The workloadinvolved in such software programming, along withthe CPU time, are essentially of the same order ofmagnitude as that for computing the entire prob-lems of (2) or (11). If one chooses instead to treat(14) implicitly by using the Lagrange multipliermethod to handle the constraint(s), and follow upwith local minimization (for inf from (6) and (12))using Newton’s algorithm, for example, extensivedomain quadratures must still be evaluated forvarious linear and nonlinear terms. This becomesoverburdensome, especially when the domain hascurvilinear boundary (because of the “daunting”work of triangulation, as we have mentioned inSec. 1). Thus, these approaches are not compati-ble with BEM, our choice of elliptic solver, to bedescribed in Sec. 2.4, which does not require exten-sive domain triangulation, and is easily adaptableto change of geometry.

Remark 2.1. The approach as suggested in Exam-ples 2.1 and 2.2 may still have some virtue: con-sequent FEM numerical error analysis seems tobe more amenable, because the problem is now a

minimization problem (see [6) or (12)] over a so-lution manifold C or M [see (7) and (13)]. Thus,there should be more numerical stability. However,the minimization problem still may have multiplesolutions in general. See Remark 2.2 below.

In the remainder of this section, we describethe iterative algorithms and numerical methods forcomputing solutions of (1).

2.1. The mountain passalgorithm (MPA)

Let E be a Banach space with norm ‖ ‖, and letJ be a C1 functional on E, with Frechet deriva-tive J ′. We say that J satisfies the Palais–Smale(PS) condition if for any sequence xn ⊆ E suchthat J(xn) bounded and J ′(xn) → 0 strongly inE′, the dual of E, then there exists a convergentsubsequence in E. The PS condition is basically acompactness condition.

Let us now state the famous Mountain–PassLemma (MPL) of Ambrosetti and Rabinowitz.

Theorem 2.1 (The Mountain-Pass Lemma[Ambrosetti & Rabinowitz, 1973]). Let E be aBanach space and let J ∈ C1(E, R) satisfy the PScondition. If there exist an e ∈ E and δ, r > 0 suchthat

(i) J(0) = J(e) = 0,(ii) r < ‖e‖ and J(x) ≥ δ > 0 ∀x ∈ Sr ≡ x ∈

E| ‖x‖ = r,then

c = infh∈Γ

maxt∈[0,1]

J(h(t)) ≥ δ (15)

is a critical value of J, where

Γ ≡ h ∈ C([0, 1], E)|h(0) = 0, h(1) = e .

By using MPL for J given in (16) below, it isnot difficult to show that (2), e.g. has a solution forΩ ∈ RN when 1 < p < p∗ = (N + 2)/(N − 2); p∗ isthe so-called critical Sobolev exponent.

The proof of MPL contains ingredients such asthe contrapositive argument from the DeformationLemma [Rabinowitz, 1986], which defies straight-forward numerical implementation. It appears thata numerical algorithm realizing the full extent of theproof of MPL for general nonlinear elliptic BVPs isquite involved and difficult, if not impossible, be-cause the saddle point stated in MPL lies in an in-finite dimensional space. Some kind of adaptation

1572 G. Chen et al.

is required in order to devise a viable algorithm.[Choi & McKenna, 1993] utilize a constructive formof MPL, an idea from [Aubin & Ekeland, 1984]. Wereword their algorithm below:

Step 1. Take an initial guess w0 ∈ E such thatw0 6= 0 and J(w0) ≤ 0, under the assumption that0 is a local minimum of J ;

Step 2. Find t∗ ∈ (0, 1) such that J(t∗w0) =maxt∈[0,1] J(tw0), and set w1 = t∗w0;

Step 3. Find the steepest descent direction v ∈ Esuch that [J(w1 + εv) − J(w1)]/ε is as negative aspossible as ε ↓ 0, obtaining v = −J ′(w1) [see (17below)]. If ‖v‖ < ε, then output and stop. Else goto the next step;

Step 4. Let λ > 0 be such that J(w1 +αv) attainsits minimum at α = λ, ∀α > 0;

Step 5. Redefine w0 =: w1 + λv. Go to Step 2.

For a semilinear elliptic BVP of the form (1),the corresponding functional J is of the form

J(v) =

∫Ω

[1

2|∇v|2 − F (x, v)

]dx ;

F (x, t) ≡∫ t

0f(x, s)ds ,

(16)

with E ≡ H10 (Ω). As noted in [Choi & McKenna,

1993], in Step 3 above, the steepest descent v =−J ′(w1) corresponds to solving a linear ellipticBVP

∆v = −∆w1 − f(x, w1) , on Ω, v|∂Ω = 0 . (17)

Assume further that J ∈ C2(E, R). The Morseindex of J at a critical point w ∈ E (cf. [Chang,1993], e.g.) is defined to be the dimension of themaximal negative definite subspace of J ′′(w). Choiand McKenna’s algorithm applies mainly to solu-tions that have Morse index 1 of the canonical func-tional J because the obtained critical point of J isthe maximum in only one direction. In certain caseswhen the underlying domain Ω of the PDE has sym-metry, their algorithm may generate solutions withMorse index 2 or higher.

An adapted version of Steps 1–5 above may befound in [Ding et al., 1999]. In this paper, ouradapted algorithm is given as follows:

Mountain Pass Algorithm (MPA)

Step 1. Choose an initial state w0 ∈ H10(Ω); set

w1 = w0.

Step 2. If

‖∆w1 + f(w1)‖L2(Ω) ≤ ε , (18)

stop and exit. Otherwise from w1, solve v:

∆v = −f(w0) on Ω, v|∂Ω = 0 . (19)

Setv = v − w1 . (20)

Then ∆v = ∆v −∆w1 = −[∆w1 + f(w1)].

Step 3. For t : T > t > 0, let λ(t) be such that1

J(λ(t)(w1 + tv)) = maxλ∈[0,1]

J(λ(w1 + tv)) .

Find t : T ≥ t ≥ 0 such that

J(λ(t)(w1 + tv)) = minT≥t≥0

J(λ(t)(w1 + tv)) .

Step 4. Update: w1 := λ(t)(w1 + tv), ∆w1 :=λ(t)(∆w1 + t∆v). Go to Step 2.

Note that solving (17) in the mountain–pass al-gorithm of [Choi & McKenna, 1993, (13)]) is iden-tical to solving (19), due to (20).

If the DO LOOP in (MPA) above stops after niterations, (18) actually say that

‖∆wn+1 + f(wn+1)‖L2(Ω) ≤ ε.The quantity

εn+1 ≡ ‖∆wn+1 + f(wn+1)‖L2(Ω) (21)

serves as an excellent indicator of how closely wn+1

satisfies ∆u+ f(u) = 0. We may thus regard εn asan absolute convergence error indicator.

Remark 2.2. Return to Example 2.2. [Ding & Ni,1986] and [Ni, 1989] have shown that under (10),we have

c = infv∈M

J(v) = infv>0

v∈H10

(Ω)

maxt≥0

J(tv) , (22)

1In our computer programs, we begin by choosing T = 1 for the first three iterations, and gradually increase T to 10 (or larger,if necessary).


where c is the smallest possible positive criticalvalue as given in the MPL. Furthermore, the twoinfima in (22) are both attained, and c is indepen-dent of the choice e in Theorem 2.1.

Our adapted algorithm MPA actually hasconsiderable feature more similar to the mini–maximization problem on the right-hand side of(22), than to the proof of the MPL itself. Thusperhaps it should be called MMA (Mini–Max Algo-rithm) instead.

The critical point as guaranteed by (22) hasthe lowest energy value J . Borrowing terminologyfrom quantum mechanics, we call this critical point(i.e. solution) a (or sometimes, the) ground state,or a least-energy solution. Other critical points,which are local minima of J(v) on the solutionmanifold M, are called local ground states in thispaper.

The very original algorithm due to [Choi &McKenna, 1993] was not devised according to (15)in the sense that they did not determine the criticalpoint by testing the inf on all paths Γ in (15). Thus,rigorously speaking, their algorithm, as well as ouradapted version, cannot be called a mountain–passalgorithm. Nevertheless, MPA here is effective infinding local ground states, with multiplicity gener-ated through varying the initial states.

The strongest result concerning the uniquenessof the ground state so far may be found in [Lin,1994], where he showed that on a convex domain,the ground state is unique.

More recently, [Li & Zhou, 1999a, 1999b] devel-oped and implemented minimax methods to com-pute multiple solutions of semilinear elliptic PDEsof higher Morse indices without any assumption ondomain symmetry.

To aid in the visual understanding we includeFigs. 1 and 2 to illustrate MPA.

2.2. The scaling iterativealgorithm (SIA)

To explain how this algorithm works, let us use thefollowing problem as a model:

∆u−au+bup = 0 , u > 0 on Ω, u|∂Ω = 0 , (23)

where a ≥ 0 and b > 0 are given constants,and p > 1. By Example 2.2 or MPL, we know

Fig. 1. The origin is located at the center of a basin, withaltitude 0. The protruding dark curve C on the right signifiesan optimal path through a mountain pass to get out of thebasin to another point, also with altitude 0. There are severalother mountain passes overlooking the depression, but theiraltitudes are higher than that of the mountain pass which iscrossed by the path C.

Fig. 2. This picture is a zoom of the right portion of Fig. 1.Note that “∗” signifies the location of the mountain pass withthe lowest altitude. This is the critical point establishedin (22). We call it a (or, sometimes, the) (global) groundstate. The location marked with “∆” is another mountainpass whose altitude is higher than “∗”’s. We call “∆” a localground state. The arrow indicates a steepest descent direc-tion. It takes several changes of descent directions in orderto reach a close vicinity of “∗”.

that (23) has at least one solution. Choose a se-quence of numbers βn > 0, n = 1, 2, . . . and definevn(x) = u(x)/βn. Then each vn is a scaling of u;

1574 G. Chen et al.

vn+1 satisfies ∆vn+1(x)− avn+1(x) + αn+1bv

pn(x) = 0 on Ω, αn+1 ≡

βpnβn+1

,

vn+1(x) > 0 on Ω,

vn+1|∂Ω = 0.

(24)

The equations in (24) suggest the following iteration algorithm:

Step 1. Choose any u0(x) ≥ 0 on Ω, u0 sufficiently smooth, u0 6≡ 0;

Step 2. Let

β0 = ‖u0‖L∞(Ω) and v0 =u0

β0,

and solve vn+1(·) and αn+1 > 0 satisfying∆vn+1(x)− avn+1(x) = −αn+1bv

pn(x) on Ω,

‖vn+1‖L∞(Ω) = 1,

vn+1|∂Ω = 0; n = 0, 1, 2, . . . ,

(25)

and let

βn+1 =βpnαn+1

, n = 0, 1, 2, . . . . (26)

Step 3. If δn+1 = ‖vn+1 − vn‖H10 (Ω) < ε, let

un+1(·) = βn+1vn+1(·) , (27)

output and stop. Else go to Step 2.

The usefulness of the above algorithm is shownin the following.

Theorem 2.2. Let Ω be a sufficiently smoothbounded domain in R2. Assume that a ≥ 0, b >0, p > 0 and p 6= 1. Choose any sufficientlysmooth v0(x) ≥ 0 on Ω, v0 6≡ 0, and let αn+1

and vn+1(·) satisfy (25) and (26). If αn|n =1, 2, . . . is bounded, then (vn+1(·), αn+1) has a con-vergent subsequence (vn+1(·), αn+1)→ (v∞(·), α∞)in C1,γ(Ω)× R for any γ : 0 < γ < 1.

Proof. The iterations in (25) and (26) can berewritten into two steps

∆wn+1 − awn+1 = −bvpn,wn+1|∂Ω = 0, n = 0, 1, 2, . . . ,

(28)and

∆vn+1 − avn+1 = −αn+1bvpn,

vn+1|∂Ω = 0, n = 0, 1, 2, . . . ,

where

vn+1 =wn+1

‖wn+1‖L∞(Ω), αn+1 =

1

‖wn+1‖L∞(Ω).

(29)

Since ‖vn‖L∞(Ω) = 1, by the standard elliptic esti-mates on (28) [Gilbarg & Trudinger, 1983], we havesome C > 0 such that

‖wn+1‖W 2,q(Ω) ≤ C , ∀ q > 1 , n = 0, 1, 2, . . . .

Therefore, the sequence αn+1 is bounded awayfrom 0, and, by the Sobolev Imbedding Theorem,wn+1 has a bounded convergent subsequence wn+1

such that wn+1 → w∞ in C1,γ(Ω) for any γ :0 < γ < 1. By assumption, αn|n = 1, 2, . . .is bounded, the sequence αn+1|n = 0, 1, 2, . . .is also bounded. Since wn+1 > 0 in Ω by thestandard maximum principle, it follows from (29)2

that w∞ > 0 in Ω. Choose a convergent sub-sequence of αn+1 and still call it αn+1, suchthat αn → α∞ > 0. From (29), we conclude thatvn+1 = αn+1wn+1 → α∞w∞ ≡ v∞ in C1,γ(Ω).

Remark 2.3.

(1) We suspect that if p > 1, then the assumptionthat αn|n = 1, 2, . . . be bounded is unneces-sary. (However, if 0 < p < 1, there is someexperimental indication that this assumption isrequired.)


(2) The iterations in (28)–(29) above define a mapT : wn+1 = Twn. In order to claim that thelimit w∞ of the subsequence wn in the proof ofTheorem 2.2 yields a solution of (23), we mustestablish that w∞ is a fixed point of the mapT . We still have not been able to achieve thisso far.

(3) By restricting p to 0 < p < p∗ ≡ (N + 2)/(N − 2), one can establish a version of Theo-rem 2.2 for Ω ⊂ RN , N ≥ 3.

Equation (25)2, or equivalently, (29)1 requiresthe search of max wn+1 on the entire Ω in or-der to determine αn+1 = 1/‖wn+1‖L∞(Ω) =1/(max wn+1). This is computationally feasiblebut inconvenient as far as implementation is con-cerned. Based upon our understanding of the prob-lem (23), as we will see from the graphics in Secs. 3and 4, we know that its solutions display the pat-tern of a spike-layer , and thus the maxima of suc-cessive iterates often occur just near the “center”x0 of some subdomain G of Ω where “Ω has themost open space”; see Theorem 4.1. This sug-gests the choice of βn = un(x0) in (24) for n =1, 2, . . . . Therefore (25)2 now becomes vn+1(x0) =un+1(x0)/un+1(x0) = 1, and we obtain the follow-ing simpler algorithm.

Scaling Iterative Algorithm (SIA)

Step 1. Choose any v0(x) ≥ 0 on Ω, v0 6≡ 0; v0

sufficiently smooth;

Step 2. Find αn+1 > 0 and vn+1(·) such that∆vn+1(x)− avn+1(x) = −αn+1bv

pn(x) on Ω,

vn+1(x0) = 1,

vn+1|∂Ω = 0;

(30)

Step 3. If

εn ≡ ‖vn+1 − vn‖H10 (Ω) < ε , (31)

output and stop. Else go to Step 2.

Note that the εn in (31) provides a relative con-vergence error indicator . Unlike the εn in (21), εnhere does not provide information as to how closelyvn+1 satisfy the equation ∆u+ f(u) = 0.

The key idea in SIA lies in condition (30)2. Byrequiring vn+1(x0) = 1, we hope to avoid, as much

as possible, the possibility that vn → 0, i.e. theconvergence to the trivial solution 0. This point-wise normalization condition (30)2 is an outgrowthof our empirical pursuit of iteration algorithms for(21) because, in our experimental trials of variouspossible designs of iterative algorithms for (23), wehave observed that the number one cause of algo-rithm failures is vn → 0 or vn →∞.

The following theorem is obvious.

Theorem 2.3. Consider the iterative algorithm(30). Let v0(·) be sufficiently regular and v0(x) ≥ 0,v0(x)>

6≡0 on Ω. Assume that (vn(·), αn) converges

to (v∞(·), α∞) in H10(Ω) ⊕ R with α∞ 6= 0. Then

u ≡ α1p−1∞ v∞ is a solution of (30).

Remark 2.4. According to our numerical experi-ence, the choice of the location of x0 ∈ Ω rarelyhas affected the limit of the converging subsequence.However, at this point we are still unable to offerany rigorous proof concerning the convergence (of asubsequence) of SIA.

Example 2.3. Here we provide some data for thecomparison of MPA and SIA. Let us leap ahead toSec. 3.5 concerning the computation of positive so-lutions of Lane–Emden’s equation ∆u+u3 = 0 withzero Dirichlet condition on the dumbbell-shaped do-main Ω7.

(i) Choosing u0(x) = RHS of (75) on Ω7 and using(76), we obtain an initial state w0 ∈ H1

0(Ω) forMPA iterations. We get a sequence which isnumerically convergent to the ground state u,with

max u = 3.562 , J = the energy (16) = 10.90 ,

εn = 10−4 , n = 7 .

(ii) Choosing the very same initial state u0(x) forSIA iterations, we also get a numerically con-vergent sequence with the same limit u, with

max u = 3.562 , J = 10.90 , εn = 10−6 ,

αn = 12.69 , n = 9 , x0 = (2, 0)

We see that MPA and SIA provide about the samenumerical efficiency and accuracy. The number of

1576 G. Chen et al.

iterations required here are, respectively, 7 and 9.In general, for most cases, only some twenty itera-tions will suffice by either MPA or SIA.

We have also tested our MPA and SIA againstthe case of a square domain in [Choi & McKenna,1993, Sec. 6], and confirmed the agreement betweenChoi and McKenna’s numerical solutions and ours.

2.3. The Direct Iteration Algorithm(DIA) and the MonotoneIteration Algorithm (MIA)

A straightforward iteration algorithm for a semilin-ear elliptic BVP (1) can be stated as follows:

Direct Iteration Algorithm (DIA) for (1)

Step 1. Choose a sufficiently smooth initial statev0(·);

Step 2. Find vn+1(·) such that∆vn+1(x) = −f(x, vn(x)), x ∈ Ω,

vn+1|∂Ω = 0;(32)

Step 3. If ‖vn+1− vn‖H10 (Ω) < ε, output and stop.

Else go to Step 2.

DIA lacks sophistication. Even for a “very stable”nonlinear ODE like

u′′(x)−u3(x)=0 on x∈ [0, 1] ; u(0)=u(1)=0 ,

(33)

[Deng et al., 1996, Example 2.1] shows that DIAmay not produce convergent solutions in general.For nearly all the nonlinear equations studied inthis paper, DIA either diverges quickly or convergesto the trivial solution 0. The only exception is thecase with sublinear nonlinearity; see Sec. 6. In thatcase, DIA actually provides an algorithm more con-venient and efficient than the others.

A different iteration scheme, with certain sim-ilarity to DIA, has been developed. It seems to beparticularly useful when a semilinear elliptic BVPhas “forcing terms” independent of u either in theequation itself or in the boundary condition. Westate the following.

Theorem 2.4 (The Monotone Iteration Scheme)[Amann, 1976; Ni, 1987; Sattinger, 1973]. LetF (x, u) be C1 with respect to (x, u) ∈ Ω × R.

Consider the boundary value problem∆u(x) + f(x, u(x)) = 0 on Ω,

Bu(x) = g(x) on ∂Ω,(34)

where Bu = u or Bu = ∂u/∂n+α(·)u, with α(x) ≥0 for all x ∈ ∂Ω, α ∈ C∞(∂Ω), and α(x) 6≡ 0 ifBu 6= u on ∂Ω, and g ∈ C2(∂Ω). Let u, v ∈ C2(Ω)satisfy u ≥ v as well as

∆u+ f(x, u(x)) ≤ 0, on Ω; Bu ≥ g(x), on ∂Ω;

∆v + f(x, v(x)) ≥ 0, on Ω; Bv ≤ g(x), on ∂Ω .

(We call u and v, respectively, a supersolution and asubsolution for (34).) Choose a number λ > 0 suchthat

λ+∂f(x, u)

∂u> 0 ∀(x, u) ∈ Ω×[v(x), u(x)] , (35)

and such that the operator (∆ − λ, B|∂Ω = 0) hasits spectrum strictly contained in the open left-halfcomplex plane. Then the mapping

T : φ 7→ w , w = Tφ , (36)

φ ∈ C2(Ω) , φ(x) ∈ [v(x), u(x)] , ∀x ∈ Ω , (37)

where w(x) is the unique solution of the BVP∆w(x)−λw(x)=−[λφ(x)+f(x, φ(x))] on Ω,

Bw(x)=g(x) on ∂Ω,

is monotone, i.e. for any φ1, φ2 satisfying (37) andφ1 ≤ φ2, we have

Tφ1 , Tφ2 satisfy (37), and Tφ1 ≤ Tφ2 on Ω .

Consequently, by letting fλ(x, u) = λu + f(x, u),the iterationsu0(x)=u(x),

(∆−λ)un+1(x)=−fλ(x, un(x)) on Ω,

n=0, 1, 2, . . . ,

Bun+1 =g on ∂Ω,

andv0(x) = v(x),

(∆− λ)vn+1(x) = −fλ(x, vn)) on Ω,

n = 0, 1, 2, . . . ,

Bvn+1 = g on ∂Ω,

yield iterates un and vn satisfying

v = v0 ≤ v1 ≤ · · · ≤ vn ≤ · · · ≤ un ≤ · · · ≤ u1 ≤ u0

= u ,


so that the limits

u∞(x) = limn→∞

un(x) , v∞(x) ≡ limn→∞

vn(x)

exist in C2(Ω). We have

(i) v∞ ≤ u∞ on Ω;(ii) u∞ and v∞ are, respectively, stable from above

and below;(iii) if v∞ 6≡ u∞, and both v∞ and u∞ are asymp-

totically stable, then there exists an unstablesolution φ ∈ C2(Ω) such that v∞ ≤ φ ≤ u∞.

Based upon Theorem 2.4, we formula the fol-lowing.

The Monotone Iteration Algorithm (MIA)

Step 1. Find a subsolution v0(·) and a supersolu-tion u0(·). Choose a λ;

Step 2. Solve the boundary value problem∆wn+1(x)−λwn+1(x)=−fλ(x, wn+1(x)) on Ω,

Bwn+1 =g on ∂Ω,

(38)

for wn+1 = vn+1 and wn+1 = un+1, respectively;

Step 3. If ‖wn+1−wn‖ < ε, output and stop. Elsego to Step 2.

The proof of Theorem 6.1 in Sec. 6 will be basedon the Monotone Iteration Scheme. BEM numer-ical analysis and computations of examples basedon MIA may be found in [Deng et al., 1996]. Finitedifference type results of MIA may be found in [Huyet al., 1986; Pao, 1987, 1992, 1995].

2.4. A boundary element numericalelliptic solver based on thesimple-layer and volume potentials

Each iterative algorithm MPA, SIA, DIA or MIArequires a numerical elliptic PDE solver. In princi-ple, the three basic numerical methods FDM, FEMand BEM should all be viable. Our choice is BEMfor the following reasons:

(i) among these three methods, BEM is the onemost readily adaptable with respect to changeof geometry, especially in 2D;

(ii) for the purpose of visualization, BEM producesthe smoothest profiles of the solution becauseit retains the special feature that solutions ofelliptic PDEs are C∞ on the domain Ω un-der the assumption that all data are C∞. (Incontrast, the smoothness of FEM solutions de-pends on the degree of piecewise polynomialsused, while FDM solutions normally displaya ragged appearance that looks definitely theleast smooth.)

(iii) The simple-layer potential representation hasa “pleasant” smoothing property (see [Chen &Zhou, 1993, Theorems 6.8.2 and 6.12.1]) mak-ing the numerical solution look smooth nearthe boundary. This is quite advantageous forthe purpose of visualization.

The basis of BEM solutions of inhomogeneouslinear elliptic BVPs is given in the following theo-rem, wherein the solution is represented as a sum ofa volume (or synonymously, Newtonian) potentialand a simple-layer potential.

Theorem 2.5. Let Ω be a bounded open domain inR2 with C∞-smooth boundary ∂Ω. Then the solu-tion of

∆u = f ∈ Hs1(Ω), s1 ≥ −1,

u|∂Ω = g ∈ Hs2(∂Ω), s2 ∈ R,(39)

can be uniquely represented as

u(x) =

∫ΩE(x− y)f(y)dy

+

∫∂Ω

E(x− y)η(y)dσy + a ∈ Hr1(Ω) ,

(40)

where E(x − y) = −(1/2π) ln |x − y|, r1 =min(s1 + 2, w2 + 3/2), and η, the unknown simple-layer density, and a, an unknown constant, can beuniquely solved by the boundary integral equations(BIEs) ∫

∂Ωη(y)dσy = 0 ,∫

∂ΩE(x− y)η(y)dσy + a

= −∫

ΩE(x− y)f(y)dy + g(x) , ∀x ∈ ∂Ω ,

(41)

with η ∈ Hr2(∂Ω), r2 = min(s1 + 1, s2 − 1).

1578 G. Chen et al.

Proof. See [Chen & Zhou, 1993, Theorems 6.3.1and 6.12.1].

Remark 2.5. If somehow it is known that thesimple-layer equation∫

∂ΩE(x− y)η(y)dσy = 0 , x ∈ ∂Ω ,

does not have a nontrivial solution η, then insteadof solving the two BIEs in (41), one can set a = 0in (40) and (41), and solve the simpler BIE∫

∂ΩE(x− y)η(y)dσy

= −∫

ΩE(x− y)f(y)dy + g(x) , ∀x ∈ ∂Ω ,

(42)

and represent the solution u of (39) as

u(x) =

∫ΩE(x− y)f(y)dy

+

∫∂Ω

E(x− y)η(y)dσy , x ∈ Ω . (43)

See [Chen & Zhou, 1993, Remark 6.12.1]. Indeed,for all domains Ωi, i = 1, 2, . . . , 9, used in the re-maining sections for computations (excluding Ω4

because it is “pathological”), only Ω1, the unit disk,requires the use of (40) and (41) with a 6= 0. Forthe rest Ωi, i 6= 1, 4, we can just use (42) and (43).

Note that ifN = 3, then the representation (43)is unique without requiring any a ∈ R as in (40),with the fundamental solution E(x) = [4π|x|]−1 be-ing used in (42) and (43).

The way to use BEM to solve the elliptic BVPs(17) and (32) in MPA and DIA, respectively, is nowclear from Theorem 2.4 and Remark 2.1. The onlycase that is not totally clear is that for (30) in SIA.Actually, this requires just a tiny amount of extrawork. Let us use (42) and (43). Return to (30), andconsider for the time being a = 0 therein. Then wefirst solve

∆v0n+1(x) = −bvpn(x) on Ω , v0

n+1|∂Ω = 0 ,

by writing

v0n+1(x) = −b

∫ΩE(x− y)vpn(y)dy

+

∫∂Ω

E(x− y)η0(y)dσy , x ∈ Ω ,

(44)

according to (42) with E(x − y) = − ln |x − y|/2π,where the unknown simple-layer density η0 is theunique solution of the BIE

∫∂Ω

E(x− y)η0(y)dσy

= b

∫ΩE(x− y)vpn(y)dy , x ∈ ∂Ω . (45)

If v0(x) > 0, then by the maximum principlev0n+1(x) > 0 on Ω. Therefore v0

n+1(x0) > 0.Define

vn+1(x)=v0n+1(x)

v0n+1(x0)

, αn+1 =[v0n+1(x0)]−1 ,

η(·)=η0(·)[v0n+1(x0)]−1 .

(46)

Then vn+1 satisfies

∆vn+1(x) = −αn+1bv

pn(x) on Ω,

vn+1(x0) = 1, vn+1|∂Ω = 0.

If a 0 in (30), we repeat (45) and (46), exceptthat now we use

E(x−y; a)

=

1

2πK0(√a|x−y|), N=2,

(K0≡the MacDonaldfunction of order 0[Abramowitz &Stegun, 1965; Chen& Zhou, 1993])

e−√a|x−y

4π|x−y| , N=3,

(47)

therein. Then vn+1 satisfies (30).We now determine the enumeration of succes-

sive errors for the BEM approach. We use (30) asan exemplar case, because the others are virtuallythe same. By (43)–(47), we have the representation

vn+1(x) = −αn+1b

∫ΩE(x− y; a)vpn(y)dy

+

∫∂Ω

E(x− y)η(y)dσy , x ∈ Ω .


Then ∆vn+1 − avn+1 = −αn+1bvpn. Similarly,

∆vn − avn = −αnbvpn−1. Therefore the error εnin (31) can be obtained by the following:

ε2n =

∫Ω

[|∇(vn+1 − vn)|2 + a|vn+1 − vn|2]dx

= ‖vn+1 − vn‖2H10 (Ω)

= −∫

Ω[∆(vn+1 − vn)

− a(vn+1 − vn)](vn+1 − vn)dx

= b

∫Ω

(αn+1vpn − αnv

pn−1)(vn+1 − vn)dx . (48)

In the evaluation of the volume potential in (44), vpnand vpn−1 have already been computed and put intostorage in the computer. We just fetch those data,substitute them into (48) and thus obtain εn+1.

The key step in the BEM approach is the nu-merical solution of the BIE (12). It is solved by dis-cretization as follows. Let Sh(∂Ω)|0 < h < h0 bea one-parameter family of finite-dimensional spacesof functions on ∂Ω. We say that Sh(∂Ω) formsan (`, m)-system with ` ≥ m + 1 and m ∈ Z+ ≡0, 1, 2, . . . in the sense of [Babuska & Aziz, 1972]if the following are satisfied:

(1) Approximating property: for each φ ∈Ht(∂Ω), there exists a φh ∈ Sh(∂Ω) such that

‖φ− φh‖Hs(∂Ω) ≤ Ct,sht−s‖φ‖Ht(∂Ω),

∀h : 0 < h < h0,

where −` ≤ s ≤ t ≤ `, |s|, |t| ≤ m, and Ct,s is apositive constant independent of h and φ.

(2) Inverse property: There exist constantsMs,t > 0 depending only on s and t such that

‖φh‖Ht(∂Ω)≤Ms,ths−t‖φh‖Hs(∂Ω), ∀φh∈Sh(∂Ω),

∀h: 0<h≤h0,

for all s ≤ t, and |s|, |t| ≤ m.

A Galerkin scheme for solving the BIE (42) is

Find ηh ∈ Sh(∂Ω) such that 〈Lηh, φk〉 = 〈G, φh〉∀φh ∈ Sh(∂Ω) ,

(49)

where L is the boundary integral operator on ∂Ωdefined by

Lη(x) =

∫∂Ω

E(x− y)η(y)dσy , ∀ ∈ ∂Ω, (50)

G is the trace on ∂Ω of the RHS of (42), andφh,i, 1 ≤ i ≤ n(h) be a basis for the linear spaceSh(∂Ω). Write

ηh =

n(h)∑i=1

aiφh,i , ai ∈ R , i = 1, 2, . . . , n(h) .

(51)Then (49) leads to the following matrix equation

Mh[a] = [b] , (52)

where Mh = [mij] is an n(h) × n(h) square matrixwith entries mij = 〈Lφh,i, φh,j〉, [a] = (ai) is ann(h)-dimensional vector with entries defined from(51), and [b] = (bi) is an n(h)-dimensional vectorwith entries bi = 〈G, φh,i〉.

It is known from [Hsiao & Wendland, 1977,1981] that the operator L is a strongly elliptic pseu-dodifferential operator with principal symbol |ξ|−1

satisfying Garding’s inequality. Therefore L is Fred-holm with a finite index. Under some accessoryconditions [Chen & Zhou, 1993, Secs. 4.6 and 4.7]L is invertible, yielding the invertibility of the ma-trix Mh for h sufficiently small, and [Ruotsalainen& Saranen, 1988, Cor. 4] show that the unique so-lution ηh of the Galerkin scheme (49) satisfies

‖η − ηh‖Hs(∂Ω) ≤ Ct,sht−s‖η‖Ht(∂Ω),

∀h: 0 < h ≤ h0,

∀s, t: s ≤ t,

−2− d ≤ s < d+ 1

2,

−3− d < t ≤ d+ 1.

(53)

If Sh(∂Ω) is a (d + 1, d)-system of the smoothestsplines of degree d, then utilizing (50) and (51), onesees that the BEM numerical solution of (39), givenby

uh(x) =

∫ΩE(x−y)f(y)dy+

∫∂Ω

E(x−y)ηh(y)dσy ,

(54)satisfies the error estimates

‖uh−u‖Hr(Ω)≤Ch2−r‖u‖H2(Ω), ∀ r: 0≤r≤2,

∀h: 0<h≤h0,

if d≥1 ,

1580 G. Chen et al.

and

‖uh−u‖Hr(Ω)≤Ch2−r‖u‖H2(Ω) ∀ r: 0≤r< 3

2,

∀h: 0<h≤h0,

if d=0 .(55)

The Galerkin scheme (49) requires work ofquadrature in order to get the entries mij and biin (52). In lieu of (49), a practical approach, calledthe collocation scheme, is much more efficient: Onechooses a set of collocation points xi|1 ≤ i ≤n(h) ⊆ ∂Ω and solves the unknown coefficientsai in (51) from

(Lηh)(xi) = G(xi) , i = 1, 2, . . . , n(h) , (56)

i.e.

n(h)∑j=1

aj

∫∂Ω

E(xi − y)φh,i(y)dσy = G(xi) ,

i = 1, 2, . . . , n(h) . (57)

With the proper choice of collocation points xiand with the use of smoothest splines (i.e. (d+1, d)-systems), Arnold and Wendland [1985] show thatthe Galerkin estimates (53) remain essentially validfor the collocation scheme (56), (57), and, hence,for the estimate (55).

In our BEM computations in the subsequentsections, quasi-uniform piecewise constant bound-ary elements are used for Sh(∂Ω). Those spacesform a (1, 0)-system in the sense of Babuska andAziz. Applying (53), with d = 0, we get

‖η − ηh‖Hs(∂Ω) ≤ Ct,sht−s‖η‖Ht(∂Ω) ,

∀h: 0 < h ≤ h0

∀ s, t: s ≤ t ,

−2 ≤ s < 1

2,

−3

2< t ≤ 1 ,

(58)

as well as (55). Our numerical experiments haveshown that our collocation scheme has convergenceestimates which are consistent with the theoreticalestimates (55) and (58).

All domains Ωi, i = 1, 2, . . . , 9, in subsequentsections are C∞, except for Ω4 in Sec. 3.3, whichis “pathological”, and Ωj , j = 7, 8, 9 in Secs. 3.5–3.7 which have two or four obtuse angular corner

points. For such nonsmooth domains, loss of reg-ularity of solutions may occur for elliptic BVPs;see [Grisvard, 1985]. However, the domains Ωj ,j = 7, 8, 9, can be modified to be C∞ by a minus-cule local smoothing of ∂Ω near the corner points inan obvious way. Indeed, with the scale of discretiza-tion adopted in our computation that is commen-surate to the memory size of our workstation (SGIExtreme Graphics 2, 256 MB RAM), we have foundno discernible difference of accuracy at all betweenΩj, j = 7, 8, 9 and the modified domains where allthe corner points have been smoothed out by minus-cule local refining using piecewise quadratic curvesegments near the corner points. Thus, for all prac-tical purposes, we may regard Ωj, j = 7, 8, 9, asC∞ domains.

Also occasional in use in this paper is FDM,when the domain and the solution has radial sym-metry ; see (72), Remarks 3.2 and and 4.1, etc. FDMsolutions also provide corroborations for, and com-parison of accuracy with, the BEM solutions. ButFDM applies to very few limited cases in this paper.

3. Graphics for Visualization of theDirichlet Problem of ∆u+ u3 = 0

The main equation, whose solutions are to bevisualized in great detail in this section, is theLane–Emden equation ∆u + u3 = 0. Other vari-ant equations, to be studied in subsequent sec-tions, have solutions with strikingly similar profiles;their graphics will be displayed only for selectedgeometries.

We have chosen four types of domains for thecomputation of numerical solutions:

(i) the unit open disk;(ii) concentric on non-concentric annuli;(iii) dumbbell-shaped domains with

varying corridor width;(iv) dumbbell-shaped domains with cavities.

(59)

The rationale for choosing (i)–(iv) is based on thespecial geometrical and topological features offeredby each type of domain(s); the disk, type (i), has thestrongest symmetry, whereupon the analytic infor-mation about the solution is also best known fromthe work of [Gidas et al., 1979, 1981]. We com-pute solutions on (i) also for the purpose of set-ting a benchmark for other researchers. By chang-ing from a disk to an annulus, i.e. to type (ii), thetopology of the domain has lost simple connected-ness and becomes multiconnected with “genus 1”,


i.e. with 1-connectivity. Topologically, its funda-mental group is isomorphic to Z [Dugundji, 1966].Rotational symmetry is destroyed after the inter-nal and external bounding circles of the annulusare made nonconcentric. Furthermore, an annulardomain is never convex.

Dumbbell-shaped domains were mentionedmuch earlier on by a few researchers (who shouldbe credited with priority, however, is unclear) as aninteresting type of domain in the study of behav-ior of solutions of BVPs. It has particular signifi-cance in mathematical biology and population dy-namics due to the compartmental feature and effectsuch domains have [Matano & Mimura, 1983]. Ourdumbbell-shaped domains are constructed by con-necting two nonidentical disks through a straight(symmetric) corridor. When the width of thecorridor is small, the dumbbell-shaped domain isnon-starshaped . As the width increases and even-tually equals the diameter of the smaller disk,the domain then becomes starshaped. Therefore,the reader can see the degeneration of dumbbell-shaped domains from being non-starshaped into astarshaped one.

Lastly, we choose dumbbell-shaped domainswith two cavities, i.e. of type (iv), for the desir-able special features that they are non-starshaped,lack any global symmetry while still maintainingsome local symmetry, and have two-connectivity.All told, nine different, mostly dissimilar domainsΩi, i = 1, 2, . . . , 9, have been selected for use hereand in subsequent sections. A total of 21 cases willbe computed and visualized in this section.

We are now in a position to treat the problem

∆u+ u3 = 0 on Ω,

u > 0 on Ω,

u|∂Ω = 0.

(60)

The corresponding energy functional of (60) is

J(v) =

∫Ω

[1

2|∇v|2 − 1

4v4]dx , v ∈ H1

0 (Ω) .

(61)

It is easy to check that for any solution u of (60),we have the energy level J(u) > 0.

3.1. The unit disk

Here Ω1 = x ∈ R2| |x| < 1. The boundary ∂Ω1 isdivided into 384 uniform panels, and the number of

Fig. 3. The unique positive solution of ∆u+ u3 = 0 on theunit disk.

Gaussian quadrature points for the domain integral(i.e. the first integral on the RHS) in (40) is 1537.

Case 3.1.a. The Unique, Radially Symmetric Pos-itive Solution [Gidas et al., 1979] on a Disk. Thisis the ground state of (60), displayed in Fig. 3, with

(SIA)max u = 3.5741 , J = 10.99 ,

ε13 = 10−6 , α13 = 12.77 , x0 = (0, 0) .

Here and in the following, (SIA) denotes the algo-rithm used, max u denotes the global approximatemaximum value of the solution, J = J(u) is the en-ergy level of the solution, ε13 denotes the relativeconvergence error from (31) and (48), α13 and x0

denote those used in (30). The choice of the initialstate v0 in Step 1 of SIA is unimportant here so wedo not need to describe it.

3.2. Nonconcentric annular domains

First, let

Ω2 = x ∈ R2| |x| < 0.9, |x−(0.2, 0)| < 0.5 . (62)

On the boundary ∂Ω2, 384 + 192 = 576 uniformpanels are used, with 192 of them placed on theinner circle of Ω2.

Case 3.2.a. The Ground State on the Nonconcen-tric Annular Domain Ω2. The ground state and its

1582 G. Chen et al.

(a)

(b)

Fig. 4. The ground state (a) and its contours (b) of (60) onthe annulus (62).

contours are displayed in Figs. 4(a) and 4(b). Wehave

(SIA)max u = 9.12 , J = 72.09 , ε16 = 10−5 ,

α16 = 80.19 , x0 = (−0.60, 0) .

Next, we change Ω2 in (62) by moving its inner

(a)

(b)

Fig. 5. The ground state (a) and its contours (b) of (60) onthe annulus (63).

boundary to be farther off-center. Let

Ω3 = x ∈ R2| |x| < 0.9 , |x− (0.35, 0)| < 0.5 .(63)

Case 3.2.b. The Ground State on the Nonconcen-tric Annular Domain Ω3. The ground state andits contours are displayed in Figs. 5(a) and 5(b).


We have

(SIA)max u = 7.38 , J = 47.43 , ε11 = 10−4 ,

α11 = 51.99 , x0 = (−0.525, 0) .

We have not been able to find any other positivesolutions on domains (62) and (63).

3.3. A “pathological” annulus,with boundary formed bytwo tangent circles

If we push the inner bounding circle in (62) and(63) to the extreme, we obtain a domain defined by

Ω4 = x ∈ R2| |x| < 0.9 , |x− (0.4, 0)| < 0.5 .(64)

In contrast to Ω2 and Ω3, Ω4 returns to being simplyconnected . Once again, 576 panels are used to dis-cretize ∂Ω4. It is “pathological” in the sense that,at the point x = (0.9, 0) on ∂Ω4, the unit exteriornormal is not well defined. There are two cusps withvertex at x. If a domain contains cusps, then thecone condition fails (at x), and many classical ellip-tic estimates may not hold. Nevertheless, numeri-cal computations of (60) by (40)–(43) can proceedwithout any trouble at all because, in the represen-tation (40) or (43), we never need to use the normalderivative ∂/∂n.

Case 3.3.a. The Ground State on the Pathologi-cal Annular Domain Ω4. The ground state is dis-played in Figs. 6(a) and 6(b). We have

(SIA)max u = 6.95 , J = 42.14 , ε12 = 10−4 ,

α12 = 47.21 , x0 = (−0.5, 0) .

3.4. The radially symmetric annulus

First, let

Ω5 = x ∈ R2|0.5 < |x| < 0.9 . (65)

A ground state of (60) on this radially symmetricannulus Ω5 is known (see [Coffman, 1984] and [Li,1990]) to be nonradially symmetric, if the inner ra-dius of the annulus increases and passes a certainpositive number. Here we will actually see that theground state looks like a hill with a single peak .Because of the symmetry which Ω5 has, any rota-tion of a given solution is again a solution. There-fore, we have a continuous one-parameter family of

(a)

(b)

Fig. 6. The ground state (a) and its contours (b) of (60) onthe pathological annulus.

ground states on Ω5, whereupon each solution isinfinitely close to a continuum of neighboring solu-tions. This unusual richness of ground states causesdifficult circumstances for numerical computation.Actually, among all the cases computed in this pa-per, this geometry constitutes the most difficult oneto deal with, with respect to numerical work. With

1584 G. Chen et al.

the use of MPA, what we have experienced is thatthe numerical iterates at first appear to be converg-ing from whatever chosen initial state, but then thetrend of convergence slows down because the iter-ates begin to “get confused” as to where they should“settle down” with respect to rotational symmetry.Small fluctuations of εn [cf. (21)] last for quite afew iterations (about 10 or so); the programmermust then realize that something is going on, askthe computer to spit out some iterates, make com-parisons and then make the decision to terminate.Otherwise the said fluctuations may persist indefi-nitely. However, with the use of SIA, this situationwill not occur because the choice of x0 in (30)2 has asymmetry-breaking effect. Numerical solutions ob-tained by SIA converge fairly fast. Their profilesshow that the peaks happen at points close to x0.

Case 3.4.a. A Ground State on the SymmetricAnnulus Ω5. It is displayed in Fig. 7, with

(SIA)max u = 13.41 , J = 162.3 , ε12 = 10−4 ,

α12 = 179.9 , x0 = (0.7, 0) .(66)

Remark 3.1. Let u be a solution of (2). It is estab-lished in [Ni, 1989] that the linearized operator

L = −[∆ + pup−1·]: v 7→ −[∆v + (pup−1)v]

has the smallest eigenvalue λ1 < 0. The next small-est eigenvalue λ2 of L satisfies λ2 ≥ 0. Note that Lcorresponds to the second derivative of J in (61). Ingeneral, if L is invertible on the appropriate func-tional space, we say that u is a nondegenerate crit-ical point .

Now, consider an annular domain

Ω = x ∈ R2|0 < a < |x| < b .

Then as [Li, 1990] has shown, when a is sufficientlyclose to b, nonradial ground states as shown in Fig. 7occur. Let u = u(v, θ) be such a ground state. Dif-ferentiating (2) with respect to θ, and using thecommutativity (∂/∂θ)∆ = ∆(∂/∂θ) in polar coor-dinates, we get

∆∂u

∂θ+ pup−1 ∂u

∂θ= 0 , on Ω ,

∂u

∂θ

∣∣∣∂Ω

= 0 .

Note that ∂u/∂θ 6≡ 0. This implies that L hasan eigenfunction ∂u/∂θ with 0 as the eigenvalue.Therefore L is not invertible and u is a degen-erate critical point of J . At a degenerate criti-cal point, the applicability of the Morse Lemma

(a)

(b)

Fig. 7. A ground state (a) and its contours (b) of (60) onthe concentric annulus (65). Any rotation of this solution isagain a solution with minimal energy.

[Chang, 1993; Mawhin & Willem, 1989] is notclear.

It is easy to see from the Implicit FunctionTheorem that any nondegenerate critical point isisolated . Case 3.4.a provides an example that theground states are all degenerate critical points andform a continuum.


Next, let us search for multipeak positive solu-tions of (60), which are again known to exist if thethickness of the symmetric annulus is sufficientlysmall. Such multipeak positive solutions are knownto be the mountain pass solutions corresponding tothe functional J on certain invariant subspaces ofthe rotational symmetry group on Ω5. (For exam-ple, a two-peak solution would be a ground stateof J on the space of functions which are invariantwith respect to rotation by 180 on Ω.) Therefore,the initial state should be chosen in those invari-ant subspaces in the hope that successive iterateswill also stay in that same invariant subspace. Ob-viously, this is a “numerically unstable” situation,since discretization and roundoff errors can accu-mulate and damage the rotational symmetry aftera large number of iterations is performed. Whatwe have observed numerically is that, after a largenumber of iterations, whether by MPA or SIA, theiterates always converge to a single-peak solutionthat is a global ground state.

On the other hand, if we perform only a smallnumber of iterations using an initial state with ro-tational symmetry of angle 2π/n, n ∈ Z+, and, say,observe a trend of numerical convergence, we let thenumerical solution data exit and terminate the it-erations. Then the solution data is expected to beclose to the n-peak positive solution (because thenumerical error accumulated after a small numberof iterations has not damaged the rotational sym-metry of angle 2π/n). In the following (Cases 3.4.b–Case 3.4.d), we use SIA to perform a small numberof iterations with initial states as indicated. Theerrors εn are comparatively larger than in the pre-ceding cases. This behavior elicits suspicion as towhether or not the numerical solution is close ornot to the true solution. Our safeguard here isthat we take several different initial states withthat same rotational symmetry and iterate fromthem. If those output multipeak solution data areall close to each other (implying the independenceof the initial states), we accept such multipeaknumerical solutions as authentic. Otherwise, wereject them.

For later use let us introduce the “mound”function

Mr0,x0(x) = cos

(π

2

|x− x0|r0

)(67)

for given r0 > 0 and x0 ∈ R2. Note thatM(r0, x0, x) = 0 for x: ‖x − x0| = r0. Let us also

introduce a rotation operator Rθ by

(Rθf)(x) = f(e−iθx) ;

e−iθx = (x1 cos θ + x2 sin θ, x2 cos θ − x1 sin θ) .

Case 3.4.b. A Two-Peak Positive Solution on theSymmetric Annulus Ω5. First, define

v0(x) =

Mr0,x0(x), r0 = 0.2, x0 = (0.7, 0),

|x− x0| ≤ r0,

0, elsewhere x ∈ Ω5,(68)

and let v0(x) = v0(x) + (Rπ v0)(x) be a period-πinitial state for SIA, resulting in

(SIA)max u = 13.60 , J = 327.0 , ε2 = 10−2 ,

α2 = 185.0 , x0 = (0.7, 0) .(69)

The two-peak solution and its contours are dis-played in Figs. 8(a) and 8(b). By comparing Fig. 7with Fig. 8 and the data in (66) with those in (69),we see that a two-peak solution is virtually a linearsuperposition of the one-peak solution with itselfbut rotated 180.

Case 3.4.c. A Three-Peak Positive Solutionon the Symmetric Annulus Ω5. Continuing fromCase 3.4.b, but using a period-2π/3 initial state u0:

v0(x) = v0(x) + (R 2π3v0)(x) + (R 4π

3v0)(x) ;

cf. v0 in (68) ;

we get

(SIA)max u = 13.60 , J = 488.6 , ε2 = 10−2 ,

α2 = 185.0, x0 = (0.7, 0) ;(70)

see Fig. 9.

Case 3.4.d. A Four-Peak Positive Solution onthe Symmetric Annulus Ω5. We use a period-π/2initial state

u0(x) =3∑k=0

(R 2kπ4v0)(x) ,

resulting in Fig. 10

(SIA)max u = 13.59 , J = 650.3 , ε2 = 10−2 ,

α2 = 184.6 , x0 = (0.7, 0) .(71)

1586 G. Chen et al.

(a)

(b)

Fig. 8. A two-peak positive solution of (60) on the concen-tric annulus Ω5.

Four is the largest number of peaks that we are ableto obtain for positive positions of (60) on Ω5.

Note that the values of J in (66)2–(71)2 ap-pear to grow linearly with respect to the number ofpeaks.

Case 3.4.e. A Radially Symmetric Positive Solu-tion on Ω5. A radially symmetric positive solution

Fig. 9. A three-peak positive solution of (60) on the con-centric annulus Ω5.

Fig. 10. A four-peak positive solution of (60) on the con-centric annulus Ω5.

of (60) does not seem to be obtainable by MPA orSIA. Iterations of arbitrarily chosen radially sym-metric initial states u0 quickly converge to a single-peak ground state in Case 3.4.a.

Here, we first assume the radial symmetry of(60) and then write it in polar coordinates:

1

r

d

dr

(rdu

dr

)+u3 =0 , u(r)|r=0.9 =u(r)|r=0.5 =0 ;

u(r)>0 , 0.5<r<0.9 .


Fig. 11. A radially symmetric positive solution of (60) onthe concentric annulus Ω5.

The nonlinear ODE above is replaced by a centeredfinite difference scheme:

u0 = un = 0 ,

ui+1 − 2ui + ui−1

h2+

1

ri

ui+1 − ui−1

2h+ u3

i = 0 ,

ui = u(0.5 + i · h) ,

ri = 0.5 + ih, h =0.4

50, i = 1, 2, . . . , n ,

n = 50 . (72)

SIA can be easily adapted for the above finite dif-ference approach. We get Fig. 11 and

(SIA)max u=9.2492 , J=1557.19 ,

ε8 =10−8 , α8 =85.1481 , x0 =(0.7, 0) .

(73)

By comparing (66)3–(73)3, we see that the radi-ally symmetric solution has an energy level J muchhigher than multipeak solutions in Cases 3.4.a–3.4.d.

Remark 3.2. On the unit disk Ω1 or the radiallysymmetric annulus Ω5, a finite difference schemelike (72) affords us a different method for validatingwhether our BEM solutions are numerically accu-rate, if the solutions have radial symmetry. FDM iscertainly the easiest to program among the variousnumerical PDE schemes. We have taken advantage

of this to confirm the accuracy of most of the ra-dially symmetric solutions below in this paper; seeCase 4.1.a, for example.

Remark 3.3. Examining the data of J in (66), (69),(70) and (71), we have found that

Jk ≈ k · (163) , k = 1, 2, 3, 4 ,

i.e., the energy Jk of the k-peak solution grows ap-proximately linearly with respect to k. This looksvery reasonable. Extrapolating, we thus obtain

J9 ≈ 9 · 163 = 1467 , J10 ≈ 10 · 163 = 1630 .

From (73)2, we see that the energy of the radi-ally symmetric solution is J = 1557.19. SinceJ9 < 1557.19 < J10, we suspect that for the annulusΩ5, the largest number of peaks a solution of (60)has is 9. Theoretical estimates in [Coffman, 1984;Li, 1990] do not furnish any information about theshape or quantity of such solutions. This is a situa-tion where numerical computation shows its value.

We suspect that for any n-peak solution in thissubsection, its Morse index is n. The radially sym-metric solution also has a finite Morse index. Wesuspect that its Morse index is m + 1, where m isthe largest number of peaks a positive solution maypossess on the annular domain.

Now, let us reduce the thickness of the annulusΩ5. We define a thinner concentric annulus

Ω6 = x ∈ R2|0.7 < |x| < 0.9 . (74)

Case 3.4.f. A Single-Peak Solution on the ThinSymmetric Annulus Ω6. This solution is displayedin Fig. 12, with

(SIA)max u = 27.11 , J = 635.2 , ε7 = 10−4 ,

α4 = 735.0 , x0 = (0.8, 0) .

Since the annulus Ω6 is thinner than Ω5, ac-cording to [Li, 1990], there will be positive solutionswith more peaks.

Case 3.4.g. A Positive Solution, with Eight Peaks,on the Thin Annulus Ω6 computed by using

u0(x)=7∑j=0

R 2πj8Mr0,x0(x) , r0 =0.1 , x0 =(0.8, 0) .

1588 G. Chen et al.

Fig. 12. A single peak, ground-state solution of (60) on theconcentric annulus Ω6.

Fig. 13. A positive solution of (60) with eight peaks on theconcentric annulus Ω6 (74).

It has

(SIA)max u = 29.15 , J = 5571 , ε4 = 10−2 ,

α4 = 247.3 , x0 = (0.8, 0) .

and is displayed in Fig. 13. Eight is the largest num-ber of peaks we are able to produce numerically.

3.5. A dumbbell-shaped domain

We consider a dumbbell-shaped domain Ω7 asshown in Fig. 14. It contains a left, smaller disk

Fig. 14. Dumbbell-shaped domain Ω7; the left disk DL hasradius 0.5 and the right DR has 1. The distance between thetwo centers of the disks is 3. The corridor has width W = 0.4.The dots on ∂Ω represent collocation points; 408 of them areplaced.

DL with radius 0.5, and a right larger disk DR withradius 1, whose centers are separated by a distanceequal to 3 along the x1-axis. A horizontal corri-dor, symmetric with respect to the x1-axis, of widthW = 0.4, is constructed to link the two disks. Inour BEM computations, ∂Ω is discretized into 408panels, and on Ω, 992 Gaussian quadrature pointsare used for integrating the volume potentials.

Case 3.5.a. The Ground State on the DumbbellΩ7. Choose

u0(x) =

−10, x ∈ DR,

0, x ∈ Ω7\DR.(75)

and solve the elliptic BVP

∆w0(x) = u0(x) , on Ω, w0|∂Ω = 0 . (76)

Obviously, w0 ∈ H10(Ω). This w0 will be used as

the initial state for MPA. Iterate by MPA. We thenobtain a single-peak positive solution as shown inFig. 15(a), with

(MPA) max u=3.562 , J=10.90 , ε7 =10−4 .

(77)

Its contours are plotted in Fig. 15(b). A carefulexamination of the contours shows that the peakpoint of the contours has moved slightly leftward(from the original center of the right disk DR), to-ward the corridor. This agrees with our intuitiveunderstanding of solutions of (60) that they prefer“open space” [Ni & Wei, 1995]).

This solution has the lowest energy. Its supportis concentrated on the right disk (i.e. the larger of


(a)

(b)

Fig. 15. The ground state (a) and its contours (b) of (60)on the dumbbell-shaped domain Ω7.

the two disks) and near the right end of the corridor.Elsewhere, u is exponentially small.

Case 3.5.b. A Local Ground State Concentratedon the Small Disk. Choose the initial state

u0(x) =

−10, x ∈ DL,

0, x ∈ Ω7\DL,(78)

obtain w0 as in (76) and iterate by MPA. We get apositive solution concentrated mainly on and nearthe small left disk DL, as displayed in Fig. 16(a),with contours shown in Fig. 16(b). This solutionhas

(MPA) max u=7.037, J(u)=42.22, ε12 =10−4 .(79)

(a)

(b)

Fig. 16. A positive solution (a) and its contours (b) of (60)concentrated on and near the small disk DL of Ω7.

Actually, if we choose any positive u0 [insteadof the u0 in (78)] supported on DL, MPA will yieldconvergence to the one in Fig. 16(a).

Case 3.5.c. A Local Ground State ConcentratedNear the Center of the Corridor. Choose the ini-tial state

u0(x) =

0, x ∈ DL ∪DR,

−10, x ∈ Ω7\(DL ∪DR),(80)

and obtain the initial state w0 from u0 by (76), anditerate by MPA. We obtain a local ground state con-centrated nearly on the center of the corridor, givenin Figs. 17(a) and 17(b), with

(MPA) max u=13.63, J(u)=159.0, ε31 =10−4 .(81)

1590 G. Chen et al.

(a)

(b)

Fig. 17. A positive solution (a) and its contours (b) of (60)concentrated near the center of the corridor of the dumbbell-shaped domain Ω7.

This “solution” has generated a considerabledebate among the authors as to whether such a so-lution could possibly be obtained analytically bythe Mountain–Pass approach. The same commentalso applies to similar situations below in Secs. 3.7,4.3 and 5.2. More investigation is certainlyrequired.

Case 3.5.d. A Two-Peak Positive Solution.Adapting an algorithm with Morse index 2 first de-veloped in [Ding et al., 1999]; see also Sec. 3.8, weare able to obtain a two-peak positive solution of(60) concentrated on both the left and right disksof Ω7, with

max u = 7.037 , J(u) = 53.12 , ε34 = 10−x4 ;

Fig. 18. A two-peak positive solution on the dumbbell-shaped domain Ω7.

Fig. 19. A starshaped domain Ω8. It is a degenerate dumb-bell. Each dot on ∂Ω represents a collocation point; thereare 408 of them.

see Fig. 18. From the data above and Figs. 15–18,we see that this solution u essentially is a combina-tion of the two solutions in Cases 3.5.a and 3.5.b. Ithas Morse index 2. As with Sec. 3.8, the purpose ofincluding Fig. 18 is to satisfy the reader’s curiosityabout the existence of multipeak positive solutions.Details of the algorithm will be presented elsewhere.

3.6. A starshaped domain degeneratedfrom a dumbbell

We expand the width W of the corridor in Ω7 fromW = 0.4 to W = 1. Then we get a new domain Ω8

as shown in Fig. 19. Even though Ω8 does not lookexactly like a star, it satisfies the starshapednesscondition, and it is no longer dumbbell-shaped. We


(a)

(b)

Fig. 20. The ground state (a) and its contours (b) on thestarshaped domain Ω8.

discretize ∂Ω8 into 408 panels, and place 992Gaussian quadrature points on Ω.

Case 3.6.a. The Ground State of (60) on the Star-shaped Domain Ω8 We choose

u0(x) =

−10, x ∈ DR, (DR is the disk with

radius 1 on theright of Ω8),

0, x ∈ Ω8\DR,

and use (76) to get the initial state w0. Iteratingwith MPA we obtain the ground state u of (60) asshown in Figs. 20(a) and 20(b),with

(MPA) max u=3.4599, J(u)=10.43, ε13 =10−1.(82)

(a)

(b)

Fig. 21. A second positive solution (a) and its contours(b) of (60) on the starshaped domain Ω8.

The reader may compare Fig. 20 against Fig. 15,and (82) against (77).

Case 3.6.b. A Local Ground State of (60) on theStarshaped Domain Ω8. Choose

u0(x)=

−10, x=(x1, x2) ∈ Ω8, x1<2−√

3

2,

0, elsewhere,

similarly as before and iterate by (MPA). We obtaina positive local mountain–pass solution as shown inFigs. 21(a) and 21(b), with

(MPA) max u=5.362, J(u)=26.18, ε13 =10−4.

(83)

1592 G. Chen et al.

Fig. 22. A dumbbell-shaped domain with two cavities, Ω9.Two circular holes are drilled on the previous dumbbell-shaped domain Ω7. The left hole is centered at (x1, x2) =(−1, 0) with radius 0.2; it maintains some local symmetry.The right hold is centered at (x1, x2) = (2, 0.3) with radius0.4; it destroys any global symmetry.

Here, we see that the two local ground states inCases 3.5.b and 3.5.c coalesce into one.

From Figs. 20(b) and 21(b), we can understandthat the positive local ground state as shown inFig. 21(a) may be “swallowed” by the global groundstate in Fig. 20(a) and, thus, disappear, if the cor-ridor portion is not long enough.

3.7. Dumbbell-shaped domains withcavities lacking symmetry

We remove two circular holes from the dumbbell-shaped domain Ω7, and obtain a dumbbell-shapeddomain with cavities, Ω9, as shown in Fig. 22. Be-cause of the placement of the hole inside the rightdisk DR, the new domain Ω9 lacks any symmetry,of a two-connectivity or with “genus 2”. Such topo-logical effects are of interest to analysts in nonlin-ear PDEs, see e.g. [Benci & Cerami, 1991]. We dis-cretize ∂Ω9 into 536 panels, and place 847 Gaussianquadrature points on Ω for volume potentials

Case 3.7.a. The Ground State (60) on Ω9. Weuse

u0(x) =

−10, for x on the right disk

with cavity,

0, elsewhere,

to get an initial state w0 ∈ H10(Ω) by (76), and

iterate with MPA, obtaining

(MPA) max u=6.003, J=33.40, ε19 =10−3 ,

and Fig. 23.

(a)

(b)

Fig. 23. The ground state (a) and its contours (b) of (60)on the dumbbell-shaped domain with cavities, Ω9.

Case 3.7.b. Local Ground States of (60) onΩ9. By making various choices of u0 (with MPA)and v0 (with SIA), we have obtained three otherlocal ground states of (60) arranged by increasingenergy level J , given in Figs. 24–26.

Obviously, there appear to exist quite a fewother positive solutions of (60) on Ω9. But we havedecided to stop at this point, and leave the pursuitto other interested researchers.

3.8. Sign-changing solutions

We return to Sec. 3.5, and use the dumbbell-shapeddomain Ω7 for the study in this subsection. Wewant to consider solutions that are not necessarily


(a)

(b)

Fig. 24. A first local ground state (a) and its contours(b) of (60) on Ω9. It is obtained by SIA, with max u = 12.83,J = 164.4, ε20 = 10−4, α20 = 79.69, x0 = (1.5, 0).

positive, i.e. find some u such that∆u+ up = 0 on Ω7,

u|∂Ω7 = 0.(84)

Even though Choi and McKenna [1993] have dis-played some sign-changing solutions of (84) on asquare, we know that in general MPA is not capa-ble of producing sign-changing solutions when thedomain Ω is not symmetric with respect to somehyperplane in RN ; see [Bartsch & Wang, 1996; Cas-tro et al., 1997; Wang, 1991; Willem, 1996]. (HereN = 2 and a hyperplane in R2 is just a line.) Asign-changing solution usually has Morse index 2 orhigher. Thus, more elaborate MMA need to be de-veloped in order to manage the higher Morse index.

(a)

(b)

Fig. 25. A second local ground state (a) and its contours(b) of (60) on Ω9. It is obtained by SIA, with max u =13, 747, J = 162.6, ε27 = 189.0, x0 = (0, 25, 0).

A useful numerical algorithm for sign-changing so-lutions of semilinear elliptic problems with Morseindex 2 was developed by [Ding et al., 1999]; it wasincorporated with FEM and yielded exemplar sign-changing solutions on triangular domains. Actu-ally, at least from an algorithmic point of view, onecan see that it is possible to generalize the ideas in[Ding et al., 1999] so that an elaborate minimaxmethod can even produce solutions of semilinearelliptic BVP with Morse index 3, 4, . . . . But suchwork requires a certain time duration for theoreticaldevelopment and numerical testing and, therefore,will be presented elsewhere, after the study has ma-tured. Here, for the sake of comprehensiveness, weinclude two examples of sign-changing solutions of

1594 G. Chen et al.

(a)

(b)

Fig. 26. A third local ground state (a) and its contours(b) of (60) on Ω9. It is obtained by SIA, with max u = 17.005,J = 361.7, ε100 = 10−4, α100 = 32.1, x0 = (2.5, 0).

(84) for p = 3. Each of them is computed by a vari-ant of the algorithm in [Ding et al., 1999] for Morseindex 2 coupled with BEM.

Case 3.8.a. A Sign-Changing Solution on the UnitDisk. We get

max u = 5.850 , J = 60.03 , ε10 = 10−3 .

This solution is displayed in Fig. 27. Note that anyrotation of this solution is again a solution.

[Dancer, 1988, p. 140] has pointed out that, fora dumbbell-shaped domain like Ω7, there should bea solution which is positive on the right disk DR andnegative on the left disk DL, and with the signs re-versed on DL and DR too, because, if u is a solutionof (84) with p = 3, then so is −u.

Fig. 27. A sign-changing solution of ∆u+ u3 = 0 with zeroboundary condition on the unit disk.

Fig. 28. A sign-changing solution of ∆u+ u3 = 0 with zeroboundary condition on the dumbbell-shaped domain Ω7.

Case 3.8.b. A Sign-Changing Solution on theDumbbell-Shaped domain Ω7. A sign-changing so-lution has been obtained as in Fig. 28; it has

max u = 3.562 , min u = −7.035 ,

J = 53.18 , ε20 = 10−4 .(85)

By comparing the data in (77), (79) and (85), wesee that this sign-changing solution in Fig. 28 is


Fig. 29. An irregular shaped domain with many compart-ments and corridors. Each “∗” indicates the likely location ofthe numerical solution of a local ground state, while “∗∗” in-dicates (“beyond any reasonable doubt”) the location of theglobal ground state because that compartment is the largest.There may exist additional local ground states that we areeither not aware of, or not able to capture numerically.

essentially a combination of the two solutions dis-played in Figs. 15(a) and 16(a), except that the partfrom Fig. 15(a) has reversed its sign. The energyvalue J in (85) is nearly equal to the sum of thevalues of J in (77) and (79).

Other profiles of sign-changing solutions (whichare “combinatorial combinations” of sorts of pairsof Figs. 15(a), 16(a) and 18(a), with just one mem-ber in the pair having sign reversed) have also beenobtained. There are five such solutions (not count-ing Fig. 27). Their graphics are omitted.

After seeing so many graphics in this section,let us now make some conclusive and inferentialcomments. Obviously, for the Lane–Emden equa-tion here and the other semilinear equations to beaddressed in the following sections, the geometry(i.e. shape), symmetry and topology of the domain

all have strong bearing on the multiplicity of posi-tive solutions. The influence of symmetry is some-what easier to understand. How about geometryand topology? The authors regard the former as amore decisive factor in generating multiple solutionsthan the latter. In an irregularly shaped domainsuch as the one shown in Fig. 29, there are many“pockets” (or “small compartments”) and “corri-dors” where local ground states of the Lane–Emdenequation thrive, such as Sec. 3.5 has obviously ledus to believe. The drilling of holes (i.e. change oftopology) on a dumbbell-shaped domain in Sec. 3.7leads to extra local ground states. However, the ac-tual effect of drilling is not necessarily in the changeof topology itself but rather, in the formation of newpockets for the extra solutions to live on. This isour observation.

4. The Singularly Perturbed DirichletProblem ε2∆u− u+ u3 = 0

The semilinear elliptic Neumann boundary valueproblem

d∆u− u+ up = 0, u > 0, on Ω, d > 0,

∂u

∂n

∣∣∣∣∂Ω

= 0,

(86)is known to be a so-called shadow system for theasymptotic state of the following model of thechemotactic aggregation stage of cellular slimemolds (amoebae) [Keller & Segel, 1970] in mathe-matical biology:

∂φ

∂t= D1∆φ− χ∇ · (φ∇ ln ψ),

∂ψ

∂t= D2∆ψ − aψ + bφ,

on Ω, t > 0; D1, D2, a, b > 0;

∂φ

∂n=∂ψ

∂n= 0 on ∂Ω, t > 0; φ(x, 0) = φ0(x), ψ(x, 0) = ψ0(x), x ∈ Ω.

(87)

See [Lin et al., 1988; Ni & Takagi, 1983, 1991]. Thecase of interest is when d is small in (86), which be-comes a singular perturbation problem. Because ofthe close relationship between the Dirichlet problemand the Neumann problem (86), [Ni & Wei, 1995]studied the singularly perturbed semilinear BVP

ε2∆u− u+ up = 0, u > 0 on Ω, ε ↓ 0,

u|∂Ω = 0,(88)

in [Ni & Wei, 1995], where ε2 corresponds to din (86). Variants of (88) have also been discussedin [Benci & Cerami, 1991; Dancer, 1988], for ex-ample. In this section, we will visualize solutionsof (88) primarily for the case p = 3 on a few se-lected domains Ωi, i = 1, 2, . . . , 9, used in Sec. 3.The visualization of the singularly perturbed Neu-mann problem (86) will be deferred to Part II [Chenet al., in preparation].

1596 G. Chen et al.

Corresponding to (88), the natural energy func-tional is defined to be

Jε(u) =1

2

∫Ω

(ε2|∇u|2 + u2)dx− 1

p+ 1

∫Ωup+1dx .

Let uε be a critical point of Jε corresponding tothe Mountain–Pass Lemma (i.e. J ′ε(uε) = 0 andJε(uε) = cε), where

cε = infh∈Γ

max0≤t≤1

Jε(h(t)) ,

and where Γ is the set of all continuous paths joiningthe origin and a fixed nonzero element e in H1

0(Ω)with e ≥ 0 and Jε(e) = 0, cf. c in (15). Then [Ni &Wei, 1995, Proposition 2.1, p. 734] cε > 0 and cε isindependent of the choice of e.

The theoretical properties of a singularly per-turbed sequence of ground states uε (88) have beenstudied at length; see [Ni & Wei, 1995]. We quotethe main result from these and state it (restrictedjust to the special case (88) in R2 here) in thefollowing.

Theorem 4.1 [[Ni & Wei, 1995, Theorem 2.2,p. 734]). Let uε be a ground state of (88). Then,for ε sufficiently small, we have

(i) uε has at most one local maximum and itis achieved at exactly one point Pε in Ω.Moreover, uε(·+ Pε)→ 0 in C1

loc(Ω− Pε\0),where Ω− Pε = x− Pε|x ∈ Ω;

(ii) d(Pε, ∂Ω) → maxP∈Ω(P, ∂Ω) as ε → 0, whered(Pε, ∂Ω) is the distance from Pε to ∂Ω.

Theorem 4.1 indicates that uε has exactly onepeak at some point Pε ∈ Ω. As ε → 0, uε → 0except at the peak Pε, thereby exhibiting a sin-gle “spike-layer”. Property (ii) says that Pεwill tend to a point P0 satisfying d(P0, ∂Ω) =maxP∈Ω d(P, ∂Ω), i.e. P0 is located near the cen-ter of some subdomain G of Ω where “Ω has themost open space”.

Remark 4.1.

(a) According to Theorem 4.1, and as confirmedby visualization from the graphics below, whenε in (88) becomes small, solutions of (88) dis-play “spikes”. Our boundary element numeri-cal method seems to capture the spike featurebetter because the numerical solution is C∞ onthe interior of the domain Ω, where the spikes

occur. If, instead, say the finite difference or fi-nite element method were used, then one likelyneeds to take adaptive measures (domain de-composition, multigrids, etc.) near the spikelocations in order to capture the special featureof their profiles.

(b) The error measure εn of the nth iterate un of(88), by (21), is

εn =

∫Ω

∣∣∣∣∆un − 1

ε2(un − upn)

∣∣∣∣2 dx1/2

.

(89)

Our numerical experience has indicated that,when ε is not small, then both MPA and SIAproduce virtually identical, equally accurate nu-merical solutions. But as ε becomes small, MPAbegins to lose accuracy. We believe that thisfact may be attributed to the numerical ill-conditionedness of the Dirichlet problem∆v = −∆w1 −

1

ε2(−w1 + w

p1), on Ω

v|∂Ω = 0,(90)

which was required in (17) of MPA. In contrast,SIA does not seem to suffer any loss of accuracy,according to our numerical experiments. We at-tribute this advantage to the scaling condition(30)2; it may have helped “normalize” the pro-file of the solution surface, at least at the pointx0. It is for this reason that most of the nu-merical solutions in this section are obtainedby SIA.

A total of 11 graphics will be presented for vi-sualization in this section.

4.1. The unit disk

We use the domain Ω1 from Sec. 3.1.

Case 4.1.a. The Unique Solution of (88) on theUnit Disk, with p = 3, and ε2 = 1, 10, 100,1000. We have obtained the following data andgraphics, as indicated in Table 1.

We may add that, when ε2 = 1 and ε2 = 10−1

in Table 1, both MPA and SIA work and producenearly equal results. However, when ε2 = 10−2 and10−3, MPA fails to work. The numerical results inTable 1 and Figs. 32 and 33 for ε2 = 10−2 and 10−3

can only be obtained by SIA.


Table 1. The data corresponding to the unique positive solution of (88) on the unit disk, with p = 3,MPA∗ at two entries mean that the numerical solutions obtained by MPA and SIA agree. But forε2 = 10−2 and 10−3, MPA does not yield convergence.

Value of ε2 max u Jε εn or εn n x0 Graphics Algorithm

1 3.951 14.71 10−4 7 Fig. 30 MPA∗

10−1 2.242 5.972 10−4 15 Fig. 31 MPA∗

10−2 2.422 5.795 10−5 46 (0, 0) Fig. 32 SIA, αn = 5.867

10−3 1.396 3.790 10−6 10 (0, 0) Fig. 33 SIA, αn = 1.948

Since the solution is radially symmetric, we can also apply an FDM to (88). Let us choose the finitedifference step size h = 1/50 in the following [which is analogous to (72)]:

1

ε2

(ui+1 − 2ui + ui−1

h2+

1

ri

ui+1 − ui−1

2h

)− ui + u3

i = 0, i = 1, 2, . . . , n− 1,

u0 = u1, un = 0, h =1

n.

Let us only list the values of max u here, obtainedfrom finite difference, for the purpose of comparisonwith Table 1:

ε2 = 1 : max u = 3.9477 ;

ε2 = 10−1 : max u = 2.2479 ;

ε2 = 10−2 : max u = 2.1546 ;

ε2 = 10−3 ; max u = 0.0001 .

(91)

We see that as ε2 ↓ 0, the difference between thedata in Table 1 from those in (91) has widened, in-dicating that measures (such as finer mesh) mustbe taken to account for the singular perturbationeffect. The singular perturbation parameter ε2 andthe mesh size h are somehow coupled in the errorbounds, as the work of [Adjerid et al., 1995] hasshown. Despite the fact that our numerical schemehas not properly adjusted the mesh size with re-spect to decreasing ε2, the graphics in this sectionhave captured the essence of the spike-layer feature.However, we hope to be able to address the issue ofcoupling between h and ε2 elsewhere in the future.

Case 4.1.b. The Unique Solution of (88) on theUnit Disk, p = 9, and ε2 = 1, 10, 100, 1000. To seehow different powers p work, we choose a mediumvalue: p = 9. The data in this case may alsoserve as benchmarks for other researchers. We havetabulated them in Table 2.

Note that all the solutions in Figs. 30–36 arethe unique mountain–pass solutions.

4.2. The radially symmetricannulus Ω6

The existence of nonradially symmetric positive so-lutions of (88) has been established in [Coffman,1984] and [Li, 1990]. Graphically, we have foundthem to be multipeak.

Case 4.2.a. A Single-Peak Ground state of (88)(p = 3). Let p = 3, ε2 = 10−2 in (88), withthe initial iterate v0(x) = Mr0,x0(x), r0 = 0.2,

Table 2. The data obtained by SIA, corresponding to the unique positive solutionsof (88) on the unit disk, with p = 9. Note that we are not able to compute the casewhen ε2 = 10−3.

Value of ε2 max u Jε εn αn n x0 Graphics

1 2.902 2.902 10−6 366.6 10 (0, 0) Fig. 34

10 1.657 2.279 4× 10−6 56.93 28 (0, 0) Fig. 35

10−2 1.489 2.204 4× 10−6 3.379 10 (0, 0) Fig. 36

1598 G. Chen et al.

Fig. 30. The ground state and the unique positive solutionof the singularly perturbed problem (88), with p = 3, ε2 = 1,on the unit disk Ω1.

Fig. 31. Same as Fig. 30, but with ε2 = 10−1.

x0 = (0.7, 0). We obtain

(SIA)max u = 2.3797, Jε = 6.320, ε55 = 10−4 ,

α55 = 5.663, x0 = (0.7, 0) ,

as displayed in Fig. 37. Note that, as above, eachrotation of u is again a solution. This solution is aground state.

We have not been able to obtain multipeak pos-itive solutions for Case 4.2.a so far.



4.3. The dumbbell-shaped domain Ω7

Case 4.3.a. Three Single-Peak, Positive SolutionsConcentrated, Respectively, on the Large Disk,Small Disk, and the Corridor. Let p = 3, ε2 =1/900 in (88). We choose three mound functionsMr0x0(x) for the initial state v0(x) in SIA iterations,and obtain the data in Table 3.

However, if we choose ε2 = 1/1000, with thesame initial states v0 as given in Table 3 and witheverything else unchanged, we obtain the data inTable 4.


Fig. 34. The ground state and the unique positive solutionof the singularly perturbed problem (88), with p = 9, ε2 = 1,on the unit disk Ω1.

Fig. 35. Same as Fig. 34, but with ε2 = 10.

The data entries for Jε = 4.060 are inconsistentwith the theory in [Ni & Wei, 1995] that the energyof the global ground state on DR should be lowerthan that of the local ground state on DL because“DR has more open space than DL”. The reasonfor this inconsistency is easy to explain: As ε2 ↓ 0,one needs to make finer discretizations of ∂Ω andΩ in order to have correspondingly higher resolutionfor the singularly perturbed problem. Otherwise nu-merical inconsistencies may occur.


Fig. 37. A single-peak, global mountain pass solution of(88), with p = 3, ε2 = 10−2, on the thin concentric annu-lus Ω6.

Note that Fig. 38 corresponds to the groundstate of (88), while Figs. 39 and 40 correspond tolocal ground states. The location of each peak foreach of the positive solutions in Figs. 38–40 hasfallen almost exactly at, respectively, the geomet-rical center of the large disk, the small disk and thecorridor. This serves as a visual confirmation of [Ni& Wei, 1995, Theorem 4.1].

1600 G. Chen et al.

Table 3. Data for three single-peak solutions of (88), with p = 3, ε2 = 1/900.

Location of thev0 = Mr0,x0

Single Peak max u Jε εn r0 x0 αn n x0 Graphics

DR (large disk) 1.251 4.01 10−6 1 (2, 0) 1.564 14 (2, 0) Fig. 38

DL (small disk) 1.801 4.145 10−6 0.5 (−1, 0) 3.242 14 (−1, 0) Fig. 39

Corridor 1.416 4.133 10−6 0.2 (0, 25, 0) 4.854 17 (0.25, 0) Fig. 40

Table 4. Data for three single-peak solutions of (88), with p = 3, ε2 = 10−3.

Location of the Single Peak max u Jε εn αn n

DR (large disk) 1.2233 4.121 10−6 1.497 14

DL (small disk) 1.7456 4.060 10−6 3.047 13

Corridor 1.3825 4.163 10−6 0.805 16

Fig. 38. A single-peak, positive solution of (88), with p = 3,ε2 = 1/900, concentrated on the large disk of the dumbbell-shaped domain Ω7.

One can also construct sign-changing solutionsfrom these three single-peak solutions just as inSec. 3.8.

Remark 4.2. The limiting peak value, max uε,p, ofa ground state uε,p in (88), as ε2 ↓ 0, of this sectionmay be obtained as follows. (The arguments weregiven implicitly in [Ni & Wei, 1995].) Consider thefollowing ODE

w′′p(r)+1

rw′p(r)−wp(r)+wpp(r)=0, 0<r<∞,

w′p(0)=0,

wp>0 on [0, ∞), limr→∞

wp(r)=0.(92)

Fig. 39. A single-peak, positive solution of (88), with p = 3,ε2 = 1/900, concentrated on the small disk of the dumbbell-shaped domain Ω7.

(If the domain Ω is in RN , then the term(1/r)w′(r), in (92) above should be replaced by((N − 1)/r)w′(r).) Equation (92) is the radiallysymmetric version of the PDE ε−2∆u− u+ up = 0in polar coordinates, where ε−2 has been “scaledout” and the angular dependence is omitted. It isknown that (92) has a unique solution satisfyingwp(0) = αp such that

limε↓0

max uε,p = αp . (93)

To obtain αp, we consider the initial value problemw′′p +1

rw′p − wp + wpp = 0, on [0, ∞),

wp(0) = β, w′p(0) = 0.(94)


Fig. 40. A single-peak, positive solution of (88), with p = 3,ε2 = 1/900, concentrated on the corridor of the dumbbell-shaped domain Ω7.

This problem has a solution wp(r; β) such that

(i) if β > αp, then there exists r0 > 0 such thatw(r0; β) = 0;

(ii) if 0 < β < αp, then limr→∞ w(r; β) = 1.

Using a method of bisection on the parameter βfor the ODE (94)1 discretized by a finite differencescheme like (72) with step size h, we obtain

(1) for p = 3,

αp,h =

2.205636, h = 10−2,

2.206171, h = 5× 10−3,

2.206193, h = 10−3;

(95)

(2) for p = 9,

αp,h =

1.810892, h = 10−2,

1.819922, h = 5× 10−3,

1.820333, h = 10−3.

(96)

The dependence of αp,h on h causes the loss of ac-curacy of αp. To obtain an accurate approximationof αp, we use a quadratic extrapolation by writing

αp,h = αp + c1(p)h+ c2(p)h2 , (97)

with three unknowns αp, c1(p) and c2(p) to be de-termined from (95) and (96), respectively. We thusobtain

αp|p=3 ≈ 2.206205 , (98)

αp|p=9 ≈ 1.820585 . (99)

The reader may use (98) and (99) to compare thevalues of max u in, respectively, Tables 1 and 2 tosee how close max u is to the “asymptotic regime.”

5. Other Variant SemilinearElliptic Dirichlet Problems

In this section, we consider three different PDEs:Henon’s equation (4), Chandrasekhar’s (5), and theLane–Emden equation (111) with p 6= 3.

5.1. Henon’s equation

Consider equation (4)1 on a general, bounded do-main Ω in R2:

∆u+ |x|`up = 0 , u > 0 , on Ω ,

u = 0 on ∂Ω , k, ` > 0 .

(100)

Since we consider only bounded domains Ω here,the growth factor |x|` should not matter very much.One might expect that solutions of (100) behave likethose of ∆u+ up = 0, other conditions being iden-tical. However, this is not true.

Case 5.1.a. Henon’s equation (100) on the UnitDisk Ω1, with ` = 1, p = 3. Even though thegoverning equation in (100) is radially symmetricon the disk Ω1, the main result in [Gidas et al.,1979, p. 221, Theorem 1′] does not apply becauseEq. (100) has explicit x-dependence. As it turnsout, some symmetry breaking occurs and a groundstate of (100) is not radially symmetric [Ni &Nussbaum, 1985]; see Figs. 41(a) and 41(b). Thisseems sort of a surprise to the novice but can beexplained roughly in just a few paragraphs given inRemark 5.1 below. We have obtained, for ` = 1,p = 3, k = 0.81,

(SIA) max u=6.075, J=35.47, ε133 =10−6 ,

α133 =17.83, x0 = (0, 0) .

(101)

We have also found a radially symmetric pos-itive solution by MPA, with Morse index 2. It isdisplayed in Fig. 42, with

(SIA)max u=5.361, J=37.09, ε11 =2×10−5,

α11 =28.74, x0 =(0, 0).

1602 G. Chen et al.

(a)

(b)

Fig. 41. A ground state (a) and its contours (b) of Henon’sequation (100) on the unit disk Ω1, with ` = 1, p = 3, andk = 0.81.

Thus it seems that, in this case, the energy of aradially symmetric positive solution is just slightlyhigher than that of a ground state.

The following is a case with a medium largepower, ` = 9. Note that |x| < 1 for all x ∈ Ω1 and,

(a)

(b)

Fig. 42. A radially symmetric positive solution (a) and itscontours (b) of Henon’s equation (100) on the unit disk, with` = 1, and p = 3. This solution has an energy level slightlyhigher than that of the ground state displayed in Fig. 41.

thus, |x|9 becomes a small number unless x is verynear to ∂Ω1.

Case 5.1.b. Henon’s equation (100) on the UnitDisk Ω1, with `=9, and p=3. Using u0(x)=−10


and w0 in (76) for the initial state of MPA, we haveobtained a ground state with the following data:

(SIA) max u = 25.23 , J = 584 , ε51 = 10−6 ,

α51 = 13.91 , x0 = (0, 0) ,(102)

and a radially symmetric solution with the followingdata

(SIA) max u = 19.76, J = 1843, ε6 = 3× 10−3,

α3 = 388.3, x0 = (0, 0).(103)

as shown in Figs. 43(a) and 43(b). One can see thatthe peak of the ground state is quite far off center,and the difference in J between the ground stateand the radially symmetric solution is much largerthan that in Case 5.1.a.

The data in both (101) and (102) may serve asbenchmarks for other researchers.

Remark 5.1. Let us explain briefly why the groundstate of Henon’s equation may lack radial symmetryon the unit disk Ω1. If a semilinear equation takesthe form

∆u+ g(r, u) = 0 , u > 0 on BR ; u|∂BR = 0 ,

(104)

where r = |x|, BR is the open ball with radiusR, and g, (∂g/∂u) are continuous with g nonin-creasing in r, then the method in [Gidas et al.,1979] applies which shows that any solution of (104)must be radially symmetric. For Henon’s equation,g(r, u) = r`up, the property of being nonincreasingin r is clearly violated.

In a way just similar to Example 2.1, one mayconsider the following two constrained variationalproblems:

Mp ≡ supu∈C

∫Ωr`|u|p+1dx ,

C ≡u ∈ H1

0 (BR)

∣∣∣∣ ∫Ω|∇u|2dx = 1

,

(105)

and

Mp,r ≡ supu∈Cr

∫Ωr`|u|p+1dx ,

Cr ≡u ∈ H1

0 (BR)

∣∣∣∣ ∫Ω|∇u|2dx = 1,

u is radially symmetric

.

(106)

(a)

(b)

Fig. 43. A ground state (a) and its contours (b) of Henon’sequation (100) on the unit disk Ω1, with ` = 9, and p = 3.

Solutions to (105) and (106) will yield solutions ofHenon’s equation after rescaling. In particular, asolution to (106) will be radially symmetric. How-ever, the distribution of the weight r`, ` > 0, heavily“favors” the part of the domain consisting of thosepoints x such that |x| = r is large, such as the an-nular strip bordering the boundary of a disk.

1604 G. Chen et al.

This, therefore, creates an effect similar to thatof the annulus case as in Sec. 3.4. Thus non-radially symmetric ground states are expectedfor (105).

Also, a consequence of this is that Mp > Mp,r

(because C ⊃ Cr).

5.2. Chandrasekhar’s equation

From (5), we consider

∆u+ 4π(u2 + 2u)3/2 = 0 , u > 0 , on Ω ;

u = 0 on ∂Ω .(107)

The energy functional of (107) is given by

J(u) =∫Ω

1

2|∇u|2 − π

[(u+ 1)(u2 + 2u)3/2 − 3

2(u+ 1)(u2 + 2u)1/2 +

3

2ln(u+ 1 +

√u2 + 2u)

]dx .

(108)

Solutions of (107) can be computed by MPA. However, because of the more involved appearance of Jin (108), we also have correspondingly much more work to do at Steps 3 and 4 of MPA. In view of this,let us consider the alternative method, SIA. Mimicking the procedure in (30) and (31), we would havederived the following iterative algorithm, with Step 2 therein replaced by Step 2′: For n = 0, 1, 2, . . . , findαn+1 ≥ 0 and vn+1(x) such that

find αn+1 ≥ 0 and vn+1(x) such that

∆vn+1(x) = −α1/2n+1 · 4π[αn+1v

2n(x) + 2vn(x)]3/2, x ∈ Ω,

vn+1(x0) = 1,

vn+1|∂Ω = 0.

(109)

As it turns out from numerical experiments we see that the adapted SIA containing (109) is divergent .After a few cases of trial-and-error, we have found that a relaxation of the unknown parameter αn by thefollowing:

1. set α0 = 1, and compute α1 by (109);

2. for n = 1, 2, 3, . . . , compute αn+1 ≥ 0 and vn+1(x) by solving

∆vn+1 = −α1/2n+1 · 4π[αn+1v

2n(x) + 2vn(x)]3/2, x ∈ Ω; αn+1 =

αn−1 + αn2

,

vn+1(x0) = 1, vn+1|∂Ω = 0,

(110)

leads to numerical convergence of both αn and vn(·)with limn→∞ αn ≡ α∞ > 0, v∞(·) = limn→∞ vn(·).Consequently, a solution u of (107) is found numer-ically, given by u(·) = α∞v∞(·).

We call the above ASIA, the adapted scalingiterative algorithm.

We have found that many of the profiles of theChandrasekhar equation are similar to those of theLane–Emden equation on Ωi, i = 1, 2, . . . , 9, inSec. 3. Note that it now makes no sense to talkabout sign-changing solutions, such as in Sec. 3.8,for the Chandrasekhar equation, because of thepower 3/2 appearing in the nonlinearity.

Case 5.2.a. Single-Peak Positive Solutions of theChandrasekhar Equation (107) on a Dumbbell-Shaped Domain, Ω7. The profiles of three single-

peak positive solutions have been obtained byASIA. Their data and graphics are indicated inTable 5.

Figure 44 represents the ground state, whileFigs. 45 and 46 represent local ground states.

5.3. The Lane–Emden equation∆u+ up = 0, p 6= 3

We consider the Lane–Emden equation with powersp 6= 3:

∆u+ up = 0 , u > 0 on Ω , u|∂Ω = 0 . (111)

Case 5.3.a. The Unique Positive Solutions of (111)on the Unit Disk Ω1, for p = 5 and p = 15. For


Table 5. The data corresponding to the three single-peak positive solutions of the Chandrasekharequation (107) on dumbbell-shaped domain Ω7.

Initial Statev0(x) = Mr0x0(x), r0 = max u J εn αn n x0 Graphics

1(DR) 0.03678 0.0006 10−6 0.1918 16 (2, 0) Fig. 44

0.5(DL) 0.3913 0.07954 10−6 0.6242 17 (−1, 0) Fig. 45

0.2 (corridor) 1.6375 2.061 2× 10−5 1.286 23 (0.25, 0) Fig. 46

Fig. 44. A single-peak ground state of the Chandrasekharequation (107) concentrated mainly on the larger disk DR.This solution can reasonably be expected to be the globalmountain–pass solution.

p = 5 and 15, using SIA we have obtained the dataand graphics indicated in Table 6.

Remark 5.2. For the unit disk Ω1, we have foundthat, for different p’s, the unique positive solu-tion in each case satisfies a monotone decreasingproperty:

up1(x) ≤ up2(x) , x ∈ Ω1 , if p1 > p2 > 1 .

(112)

We still have not been able to prove (112). How-ever, we must remark that (112) relies on the factthat Ω1 ⊂ R2; it will no longer hold for the unit ballin R3, due to the presence of the critical exponentp∗ = (N + 2)/(N − 2).

Remark 5.3. A recent paper by Ren and Wei [1996]gives a sharp characterization of the asymptotic

Fig. 45. A single-peak positive solution of the Chan-drasekhar equation (107) concentrated mainly on the smallerdisk DL.

behavior of the ground states of the Lane–Emdenequation on a 2D smooth domain Ω when the ex-ponent p grows large. Let up be a ground state of(111). Then [Ren & Wei, 1996, Theorem 1.4] showsthat

1≤ limp→∞

‖up‖L∞(Ω)≤ limp→∞

‖up‖L∞(Ω)≤√e .

(113)

In Table 6, we have max u15 = 1.5469, which differsfrom

√e ≈ 1.6487 with about 6% of relative devi-

ation. (This deviation is not necessarily an errorbecause it is not known so far whether (113) givesthe tightest estimates.) As p increases past 15, wehave found that max up decreases in value, some-how indicating that a finer discretization is calledfor in order to improve accuracy. After p passes20, computer arithmetic overflow occurs, prevent-ing further computations.

1606 G. Chen et al.

Table 6. The data corresponding to the unique positive solutions of the Lane–Emdenequation (111) on the unit disk Ω1, for p = 5 and 15.

p max u J εn αn n x0 Graphics

5 1.3307 4.831 10−6 29.51 21 (0, 0) Fig. 47

15 1.5469 1.617 2.5× 10−5 449.1 8 (0, 0) Fig. 48

Fig. 46. A single-peak positive solution of the Chan-drasekhar equation (107) concentrated mainly on thecorridor.

Fig. 47. The unique ground state of (111) on the unit diskΩ1, with p = 5.

Fig. 48. The unique ground state of (111) on the unit diskΩ1, with p = 15.

Ren and Wei [1996, Theorem 1.3] have furthershown that as p → ∞, for a subsequence pn of p,we have

upnpn∫Ωupnpn(x)dx

→ δ(x0) (114)

in the sense of distribution for some unique pointx0 ∈ Ω, called the “blow-up” or “condensation”point, where x0 can be characterized as a criticalpoint of the function φ(x) ≡ g(x, x), where g(x, y)is the “regular part” of the Green’s function G(x, y)for the domain Ω:

∆xG(x, y) = −δ(x− y) , x, y ∈ Ω ;

G(x, y)|x∈∂Ω = 0 , y ∈ Ω .

Furthermore, if Ω is convex, then (114) holds forthe entire sequence p. Thus, as p grows large, theground states “look more and more like a singlespike”.


We nevertheless wish to reiterate that for theLane–Emden equation (2), even though it is be-lieved to be true that its ground states all have asingle peak, there seems to be no rigorous proofso far.

6. Sublinear Dirichlet Problem∆u+ up = 0, 0 < p < 1

Let us consider

∆u+ up = 0 , u > 0 on Ω ,

u|∂Ω = 0 , 0 < p < 1 .(115)

With the sublinear growth of nonlinearity up in(115), we have a complete grasp of existence anduniqueness of the positive solution.

Theorem 6.1. Equation (115) has a uniquesolution.

Its proof, involving the Monotone IterationScheme, Theorem 2.4, may be found in [Ni, 1987,pp. 69–70], for example.

6.1. Solutions of (115) bydirect iteration

Obviously, MIA appears to be “the algorithm” to beused for (115), since the proof of Theorem 6.1 wascarried out that way. Our numerical experience haseasily confirmed this. MPA does not work for (115)because p < 1. We have experimented with SIA,which also converges nicely, and the numerical so-lutions coincide with those obtained by MIA. In ournumerical study, however, we have also found thatthe use of DIA for (115) suffices. This fact is givenin the following.

Theorem 6.2 [Convergence of the Direct IterationAlgorithm for (115)]. Let Ω be a bounded opendomain in RN , with C2,δ-smooth boundary ∂Ω forsome 0 < δ < 1. Let an initial state u0(·) be suf-ficiently smooth, u0(x) ≥ 0 on Ω, and u0 6≡ 0. Letun+1 be the solution of

∆un+1+upn=0, on Ω, for given p: 0<p<1,

un+1|∂Ω =0.(116)

Then un converges to the unique solution u of (115)in C2(Ω).

Proof. Let G(x, y) be the Green’s functionsatisfying

∆G(x, y)=−δ(x−y), ∀x, y∈Ω,

G(x, y)=0, ∀ y ∈ ∂Ω, for each x∈Ω,x 6=y.

Then it is known that G(x, y) ≥ 0 almost every-where on Ω× Ω. We have the representation

un+1(x) =

∫ΩG(x, y)upn(y)dy , ∀x ∈ Ω . (117)

Let ρ(·) be the solution of

∆ρ+ 1 = 0 , on Ω , ρ|∂Ω = 0 . (118)

Then from the fact that G ≥ 0, along with (117)and (118), we get

‖un+1‖C0 ≤ ‖upn‖C0 ·maxx∈Ω

∣∣∣∣ ∫ΩG(x, y)dy

∣∣∣∣= ‖un‖pC0‖ρ‖C0 , where C0 is the

C(Ω)-norm.

(119)

From the maximum principle, we have un(x) > 0on Ω for n ≥ 1. Choose any open subdomain Ω′

such that Ω′ ⊂ Ω. Then

un+1(x) ≥∫

Ω′G(x, y)upn(y)dy

≥(

minΩ′

un

)p ∫ω′G(x, y)dy

≥(

minΩ′

un

)pD(Ω′) , ∀x ∈ Ω′ , (120)

where C(Ω′) ≡ minx∈Ω

′∫Ω′ G(x, y)dy > 0. From

(119) and (120), therefore, we have

C(Ω′) ·(

minΩ′

un

)p≤ min

Ω′un+1 ≤ ‖un+1‖C0

≤ ‖un‖pC0‖ρ‖C0 . (121)

Since 0 < p < 1, 1−p > 0, we can choose a positivenumber M > 1 so large that ‖u0‖L∞(Ω) ≤ M and

‖ρ‖C0 ≤M1−p. Then, from (121),

‖u1‖C0 ≤ ‖u0‖pL∞(Ω)‖ρ‖C0 ≤MpM1−p = M .

Similarly,

‖ui‖C0 ≤M , for i = 2, 3, . . . . (122)

From the standard elliptic estimate for (116)

1608 G. Chen et al.

[Gilbarg & Trudinger, 1983 Chap. 7], we have

‖un+1‖W 2,q(Ω) ≤ C1‖un‖pLq(Ω) ≤ C2‖un‖pL∞(Ω)

≤ C3Mp , by (122) .

Choose q sufficiently large. By the Sobolev Imbed-ding Theorem, we have, for some α: 0 < α < 1,

‖un+1‖C1,α(Ω) ≤ C4Mp ≤ C4M ,

and thus, by integrating d(upn+1), the differential ofupn+1, from a point on ∂Ω, we get

‖upn+1‖Cα(Ω) ≤ C5(M) , independent of n .

By the Schauder estimates,

‖un+2‖C2,α(Ω)≤C6(M) , independent of n,

by [Gilbarg & Trudinger,

1983, Chap. 6] .

Therefore, the sequence un is bounded in C2,α(Ω)and, therefore it contains a convergent subsequencein C2(Ω) with limit u. This u satisfies u ≥ 0 and

∆u+ up = 0 , on Ω , u|∂Ω = 0 .

We now show that u 6≡ 0.Choose m > 0 so small that m = C(Ω′)

11−p .

cf. (121). We then have, if an = minΩ′ un ≤ m forsome n, then, from (120),

an+1 ≡ minΩ′

un+1 ≥ C(Ω′)(

minΩ′

un

)p= C(Ω′)apn = m1−papn

≥ a1−pn apn = an .

This implies that the sequence an must be in-creasing below the level m. On the other hand, ifan ≥ m, then

an=1 ≥ C(Ω′)apn = m1−p · apn

≥ m1−pmp = m,

i.e. once the sequence an gets above the level m,it must stay above the level m.

The above guarantees that

minΩ′

un ≡ an ≥ mina0, m > 0 ,

for a subdomain Ω′ such that v0 > a0 > 0 almosteverywhere on Ω′. Hence u, as the limit of un,cannot be identically 0. Therefore, u must be theunique solution of u of (115).

Note that the entire sequence un convergesto u, because every subsequence of un does.

We now provide some examples obtained byDIA.

Case 6.1.a. The Unique Solution of (115) on theUnit Disk Ω1, for p = 1/3 and 2/3. Using DIA,we have obtained the following data and graphicsin Table 7.

Case 6.1.b. The Unique Solution of (115) on theDumbbell-Shaped Domain Ω7. The data are givenin Table 8.

Table 7. The data corresponding to the unique solution of (115) on theunit disk Ω1, for p = 1/3, 2/3.

p max u J εn n Graphics

1/3 0.1046 −1.562 × 10−2 10−6 11 Fig. 49

2/3 0.00756 −2.998 × 10−5 10−6 24 Fig. 50

Table 8. The data corresponding to the unique solution of (115) on thedumbbell-shaped domain Ω7.

p max u J εn n Graphics

1/3 1.058 × 10−1 −1.641 × 10−2 10−6 12 Fig. 51

2/3 7.735 × 10−3 −3.162 × 10−5 10−6 24 Fig. 52


6.2. A consequence of visualization:Monotonicity of solutions of(115) with respect to p

In comparing the data for the solutions of (115),we have found an interesting phenomenon: solu-tions of (115) decrease pointwise with respect to thepower p, i.e. let up denote the solution of (115) cor-responding to p, 0 < p < 1. Then up1(x) < up2(x)if 0 < p2 < p1 < 1, for all x ∈ Ω. The telltale signof this can be visualized from Figs. 49–52. Can we

Fig. 49. The unique solution of (115) on the unit disk,p = 1/3.

Fig. 50. The unique solution of (115) on the unit disk,p = 2/3.

Fig. 51. The unique solution of (115) on the dumbbell-shaped domain Ω7, p = 1/3.

Fig. 52. The unique solution of (115) on the dumbbell-shaped domain Ω7, p = 2/3.

establish a rigorous proof of this? The following isa theoretical outcome of visualization.

Theorem 6.3. Let Ω ⊆ S, where S is a strip withwidth at most

√2: S = (x1, x2) ∈ R2||x2| <

√2.

Then up ↓ pointwise on Ω as p ↑, for 0 < p < 1.

We first show that the following lemmaholds.

1610 G. Chen et al.

Lemma 6.1. We have 0 < up(x) < 1 on Ω, ifΩ ⊆ S.

Proof. Set φ(x) = 1− (1/2)x22. We have 0 ≤ ϕ ≤ 1

on S, and

∆φ+ φp = −1 + φp ≤ 0 on S .

Since φ ≥ 0 on ∂Ω, φ is a supersolution of (115),for all p: 0 < p < 1. A subsolution ψ with ψ ≤ φcan be constructed (in the “usual” way) as follows.Set ψ = εψ1 for some ε > 0, where ψ1 is the firsteigenfunction of ∆, characterized by the followingproperties:

∆ψ1 + λ1ψ1 = 0 , ψ1 > 0 on Ω ,

ψ1| − ∂Ω = 0 , λ1 > 0 .

(We may normalize ψ1 by ‖ψ1‖L2(Ω) = 1.) We thenhave

∆ψ + ψp = ε∆ψ1 + εpψp1 = εpψp1 − ελ1ψ1

= εψ1(εp−1ψp−11 − λ1) > 0

if and only if

(εψ1)p−1 > λ1 , or1

λ1> ε1−pψ1−p

1 . (123)

Since 1−p > 0 and ψ1 is bounded, (123) can alwaysbe achieved by taking ε small. Now, choosing ε stillsmaller if necessary, we have ψ < φ. By the Mono-tone Iteration Scheme, Theorem 2.4, we concludethat there exists a solution vp of (115) for this p,0 < p < 1, such that ψ ≤ vp ≤ φ. Since (115) hasup as the unique solution, we must have up ≡ vp.Therefore 0 ≤ ψ ≤ up ≤ φ ≤ 1. We then haveup < 1 on Ω since φ is a strict supersolution.

We can now complete the proof.

Proof of Theorem 6.3. First, we note that, if q >p > 0, then up is a supersolution for Eq. (115) cor-responding to q:

∆up + uqp = −upp + uqp = upp(uq−pp − 1) < 0 ,

by Lemma 6.1, since q − p > 0. Let ψ1 be the firsteigenfunction of ∆ as in the proof of Lemma 6.1and choose ε > 0 so small that ψ = εψ1 is againa subsolution of (115) corresponding to q. Thenψ = εψ1 < up on Ω by the Hopf Boundary PointLemma. Again, by the uniqueness of uq, we have

0 < ψ = εψ1 < uq < up < 1 on Ω, for all1 > q > p > 0. The proof is complete.

All the domains Ωi, i = 1, 2, . . . , 9, in this pa-per satisfies the strip-width condition Ωi ⊆ S. Webelieve Theorem 6.3 holds for any bounded open do-main Ω, but so far we have not been able to providea proof.

Acknowledgments

The authors wish to thank Professors Kung-ChingChang, P. Joe McKenna, and Zhonghai Ding forhelpful discussions.

ReferencesAbramowitz, M. & Stegun, I. A. [1965] Handbook of

Mathematical Functions (Dover, NY).Adams, R. A. [1975] Sobolev Spaces (Academic Press,

NY).Adjerid, S., Aiffa, M. & Flaherty, J. E. [1995] “High-order

finite element methods for singular perturbed ellipticand parabolic problems,” SIAM J. Appl. Math. 55,520–543.

Amann, H. [1976] “Supersolution, monotone iterationand stability,” J. Diff. Eq. 21, 367–377.

Amann, H. & Crandall, M. G. [1978] “On some exis-tence theorems for semilinear elliptic equations,” In-diana Univ. Math. J. 27, 779–790.

Ambrosetti, A. & Rabinowitz, P. H. [1973] “Dual vari-ational methods in critical point theory and applica-tions,” J. Funct. Anal. 14, 349–381.

Aris, R. [1975] The Mathematical Theory of Diffusionand Reaction in Permeable Catalysts, Vol. I (OxfordUniv. Press, Oxford, UK).

Arnold, D. & Wendland, W. [1985] “The convergenceof spline collocation for strongly elliptic equations oncurves,” Numer. Math. 47, 317–341.

Aubin, J. P. & Ekeland, I. [1984] Applied NonlinearAnalysis (Wiley, NY).

Babuska, I. & Aziz, A. [1972] The Mathematical Founda-tion of the Finite Element Method with Applications toPartial Differential Equations (Academic Press, NY).

Bartsch, T. & Wang, Z. Q. [1996] “On the existenceof sign-changing solutions for semilinear Dirichletproblems,” Topol. Meth. Nonlin. Anal. 7, 115–131.

Benci, V. & Cerami, G. [1991] “The effect of the domaintopology on the number of positive solutions of non-linear elliptic problems,” Arch. Rat. Mech. Anal. 114,79–94.

Bers, L. [1953] “On mildly nonlinear partial differen-tial equations of elliptic type,” J. Res. Nat. Bureauof Standards 51, 229–236.


Bieberbach, L. [1916] “∆u = eu und die automorphenfunctionen,” Mathematische Annalen 77, 173–212.

Bramble, J. H. [1993] Multigrid Methods , PitmanResearch Notes in Mathematics, Vol. 294 (JohnWiley).

Brenner, S. C. & Scott, R. L. [1994] The MathematicalTheory of Finite Element Methods (Springer-Verlag,NY).

Castro, A., Cossio, J. & Neuberger, J. M. [1997] “A sign-changing solution for a superlinear Dirichlet problem,”Rocky Mountain J. Math. 351, 1919–1945.

Chandrasekhar, S. [1939] An Introduction to the Studyof Stellar Structure (University of Chicago Press).

Chang, K. C. [1993] Infinite Dimensional Morse Theoryand Multiple Solution Problems (Birkhauser, Boston).

Chen, G. & Zhou, J. [1993] Boundary Element Methods(Academic Press, London–San Diego).

Chen, G. & Zhou, J. [1993] Vibration and Damping inDistributed Systems, WKB and Wave Methods, Visu-alization and Experimentation: Vol. II (CRC Press,Boca Raton, FL).

Chen, G., Deng, Y., Ni, W. M. & Zhou, J. [1999] “Bound-ary element monotone iteration scheme for semilinearelliptic partial differential equations, Part II: Quasi-monotone iteration for coupled 2× 2 systems,” Math.Comput. 69, 629–652.

Chen, G., Deng, Y., Ni, W. M. & Zhou, J. [in review]“Ibid, Part III: Interaction of spatial and boundarynonlinearities.”

Chen, G., Morris, P. J. & Zhou, J. [1994] “Visualizationof special eigenmode shapes of a vibrating ellipticalmembrane,” SIAM Rev. 36, 453–469.

Chen, G., Ni, W. M. & Zhou, J. [in preparation] “Visu-alization for solutions of semilinear elliptic equationswith Neumann and Robin boundary conditions.”

Chen, Y. & McKenna, P. J. [1997] “Numerical variationalmethods for approximating travelling waves in a non-linearly suspended beam,” J. Diff. Eq. 136, 325–355.

Choi, Y. S. & McKenna, P. J. [1993] “A mountain passmethod for the numerical solution of semilinear ellip-tic problems,” Nonlin. Anal. 20, 417–437.

Choi, Y. S. & McKenna, P. J. [1996] “Numericalmountain pass methods for nonlinear boundary valueproblems,” Nonlinear Problems in Applied Mathemat-ics eds. Angell, T. S. et al. (SIAM, Philadelphia),pp. 86–95.

Ciarlet, P. G. [1978] The Finite Element Method forElliptic Problems (North-Holland, Amsterdam–NY–Oxford).

Ciarlet, P. G., Schultz, M. H. & Varga, R. S. [1967]“Numerical methods of high-accuracy for nonlinearboundary value problems,” Numer. Math. 9, 394–430.

Coffman, C. V. [1984] “A nonlinear boundary value prob-lem with many positive solutions,” J. Diff. Eq. 54,429–437.

Dancer, E. N. [1988] “The effect of domain shape on

the number of positive solutions of certain nonlinearequations,” J. Diff. Eq. 74, 120–156.

Deng, Y., Chen, G., Ni, W. M. & Zhou, J. [1996] “Bound-ary element monotone iteration scheme for semilinearelliptic partial differential equations,” Math. Comput.65, 943–982.

Ding, W. Y. & Ni, W. M. [1986] “On the existence of pos-itive entire solutions of a semilinear elliptic equation,”Arch. Rat. Mech. Anal. 91, 283–308.

Ding, Z., Costa, D. & Chen, G. [1999] “A high linkingmethod for sign changing solutions of semilinear ellip-tic equations,” Nonlin. Anal. 38, 151–172.

Douglas, J. & Dupont, T. [1975] “A Galerkin methodfor a nonlinear Dirichlet problem,” Math. Comput. 29,689–696.

Dugundji, J. [1966] Topology (Allyn and Bacon, Boston).Eydeland, A. & Spruck, J. [1988] “The inverse power

method for semilinear elliptic equations,” NonlinearDiffusion Equations and Their Equilibrium States I ,MSRI Publications, Vol. 12, eds. Ni, W. M. et al.(Springer-Verlag, NY).

Fowler, R. H. [1931] “Further studies on Emden’s andsimilar differential equations,” Quart. J. Math. 2,259–288.

Geirer, A. & Meinhardt, H. [1972] “A theory of biologicalpattern formation,” Kybernetic 12, 30–39.

Gilbarg, D. & Trudinger, N. S. [1983] Elliptic PartialDifferential Equations of Second Order, 2nd edition(Springer-Verlag, Berlin).

Gidas, B., Ni, W. M. & Nirenberg, L. [1979] “Symmetryand related properties via the maximum principle,”Commun. Math. Phys. 68, 209–243.

Gidas, B., Ni, W. M. & Nirenberg, L. [1981] “Symmetryof positive solutions of nonlinear elliptic equations inRn,” Advances in Math., Supplementary Studies A7,369–402.

Greenspan, D. & Parter, S. V. [1965] “Mildly nonlinearelliptic partial differential equations and their numer-ical solution, II,” Numer. Math. 7, 129–146.

Grisvard, P. [1985] Elliptic Problems in NonsmoothDomains (Pitman Advanced Publishing Program,Boston).

Henon, M. [1973] “Numerical experiments on the stabil-ity of spherical stellar systems,” Astro. Astrophys. 24,229–238.

Hill, S. [1997] “Numerical and theoretical investigation ofthe variational formulation of a water wave problem,”PhD Dissertation, Math. Dept., Univ. of Connecticut,Storrs, CT.

Hsiao, G. & Wendland, W. L. [1977] “A finite elementmethod for some integral equations of the first kind,”J. Math. Anal. Appl. 58, 449–481.

Hsiao, G. & Wendland, W. L. [1981] “The Aubin–Nitschelemma for integral equations,” J. Integral Eq. 3,299–315.

Humphreys, L. D. & McKenna, P. J. [1999] “Multiple

1612 G. Chen et al.

periodic solutions for a nonlinear suspension bridgeequations,” IMA J. Appl. Math. 63, 37–49.

Huy, C. U., McKenna, P. J. & Walter, W. [1986] “Finitedifference approximations to the Dirichlet problem forelliptic systems,” Numer. Math. 49, 227–237.

Ishihara, K. [1984] “Monotone explicit iterations of the fi-nite element approximations for the nonlinear bound-ary value problems,” Numer. Math. 45, 419–437.

Keller, E. F. & Segel, S. A. [1970] “Initiation of shinemode aggregation viewed as an instability,” J. Theo.Biol. 26, 399–415.

Keller, H. B. [1968] Numerical Solutions for Two-PointBoundary-Value Problems , Blaisdell (Waltham, Mas-sachusetts).

Li, Y. Y. [1990] “Existence of many positive solutions ofsemilinear elliptic equations on annulus,” J. Diff. Eq.83, 348–367.

Li, Y. & Zhou, J. [in review] “A minimax method forfinding critical points with a general Morse index andits applications to semilinear PDE.”

Li, Y. & Zhou, J. [in review] “Convergence and Morse in-dex results on a minimax method for finding multiplecritical points.”

Lieb, E. & Yau, H.-T. [1987] “The Chandrasekhar theoryof stellar collapse as the limit of quantum mechanics,”Commun. Math. Phys. 112, 147–174.

Lin, C. S. [1994] “Uniqueness of solutions minimiz-ing the functional

∫Ω|∇u|2/(

∫Ω|u|p+1)2/(p+1) in Rn,”

Manuscripta Mathematica 84, 13–19.Lin, C. S., Ni, W. M. & Takagi, I. [1988] “Large ampli-

tude solutions to a chemotaxis system,” J. Diff. Eq.72, 1–27.

Lions, L. J. [1969] Quelques Methodes de Resolution desProblemes aux Limites Non Lineaires (Dunon, Paris).

Lions, J. L. & Magenes, E. [1972] Non-homogeneousBoundary Value Problems and Applications, Vol. I(Springer-Verlag, NY).

Matano, H. & Mimura, M. [1983] “Pattern formation incompetition diffusion systems in nonconvex domains,”Publ. RIMS, Kyoto Univ. 19, 1049–1079.

Mawhin, J. & Willem, M. [1989] Critical Point Theoryand Hamiltonian Systems (Springer-Verlag, NY).

Nehari, Z. [1960] “On a class of nonlinear second-orderdifferential equations,” Trans. Amer. Math. Soc. 95,101–123.

Ni, W. M. [1987] Some Aspects of Semilinear Ellip-tic Equations (Dept. of Math. National Tsing HuaUniv., Hsinchu, Taiwan, R.O.C.).

Ni, W. M. [1989] “Recent progress in semilinear ellipticequations,” RIMS Kokyuroku 679 (Kyoto University,Kyoto, Japan), pp. 1-39.

Ni, W. M. [1998] “Diffusion, cross diffusion and theirspike-layer steady states,” Notices of Amer. Math.Soc. 45, 9–18.

Ni, W. M. & Nussbaum, R. [1985] “Uniqueness and

nonuniqueness for positive radial solutions of ∆u +f(u, r) = 0,” Commun. Pure Appl. Math. 38, 67–108.

Ni, W. M. & Takagi, I. [1991] “On the shape of leastenergy solutions to a semilinear Neumann problem,”Commun. Pure Appl. Math. 44, 819–851.

Ni, W. M. & Takagi, I. [1983] “Locating the peaks of leastenergy solutions to a semilinear Neumann problem,”Duke Math. J. 70, 247–281.

Ni, W. M. & Wei, J. [1995] “On the location and profile ofspike-layer solutions to singularly perturbed semilin-ear Dirichlet problems,” Commun. Pure Appl. Math.48, 731–768.

Pao, C. V. [1987] “Numerical solutions for some cou-pled systems of nonlinear boundary value problems,”Numer. Math. 51, 381–394.

Pao, C. V. [1992] Nonlinear Parabolic and EllipticEquations (Plenum Press, NY).

Pao, C. V. [1995] “Block monotone iterative methods fornumerical solutions of nonlinear elliptic equations,”Numer. Math. 72, 239–262.

Parter, S. V. [1965] “Mildly nonlinear elliptic partial dif-ferential equations and their numerical solutions I,”Numer. Math. 7, 113–128.

Protter, M. & Weinberger, H. [1967] Maximum Princi-ples in Differential Equations (Prentice-Hall, Engle-wood Cliffs, NJ).

Rabinowitz, P. [1986] Minimax Method in Critical PointTheory with Applications to Differential Equations,CBMS Regional Conf. Series in Math., No. 65, AMS,Providence.

Ren, X. & Wei, J. [1996] “Single-point condensation andleast energy solutions,” Proc. Am. Math. Soc. 124,111–120.

Ruotsalainen, K. & Saranen, I. [1988] “On the conver-gence of the Galerkin method for nonsmooth solutionsof integral equations,” Numer. Math. 54, 295–302.

Sakakihara, J. [1987] “An iterative boundary inte-gral equation method for mildly nonlinear ellip-tic partial differential equations,” Boundary Ele-ments IX ed. Brebbia, C. A. (Springer-Verlag, NY),pp. 13.49–13.58.

Sattinger, D. [1971/72] “Monotone methods in nonlin-ear elliptic and parabolic boundary value problems,”Indiana Univ. Math. J. 21, 979–1000.

Sattinger, D. [1973] Topics in Stability and Bifurca-tion Theory, Lecture Notes in Mathematics #309(Springer-Verlag, Berlin–Heidelberg–NY).

Strang, G. & Fix, G. J. [1973] An Analysis of theFinite Element Method (Prentice-Hall, EnglewoodCliffs, NJ).

Wang, Z. Q. [1991] “On a superlinear elliptic equation,”Ann. Inst. Henri Poincare 8, 43–57.

Willem, M. [1996] Minimax Theorems (Birkhauser,Boston).

algorithms and visualization for solutions of nonlinear ...j.zhou/cnz.pdf · algorithms and...

Documents