on the structure of the positive solutions of the logistic equation with nonlinear diffusion

Journal of Mathematical Analysis and Applications 268, 200–216 (2002)doi:10.1006/jmaa.2001.7815, available online at http://www.idealibrary.com on

On the Structure of the Positive Solutions of theLogistic Equation with Nonlinear Diffusion

Manuel Delgado and Antonio Suarez1

Departamento de Ecuaciones Diferenciales y Analisis Numerico,Facultad de Matematicas, University of Seville,

c/ Tarfia s/n, 41012 Seville, SpainE-mail: [email protected]; [email protected]

Submitted by Chia Ven Pao

Received January 3, 2001

1. INTRODUCTION

In this work we study the structure of the positive solutions of thedegenerate logistic equation, i.e., of the elliptic boundary value problem

d�wm=σ w − b�x�wr in �

w= 0 on ∂�(1)

where � is a bounded domain of �NN ≥ 1, with a smooth boundary∂�� is a general second-order uniformly elliptic operator; b is a positivefunction; m ≥ 1; r > 1; d is a positive constant; and σ is a real parameter.Equation (1) was introduced in biological models by Gurtin and MacCamy[7] (see also [13] and [14]) who were describing the dynamics of biologicalpopulations whose mobility is density dependent. In (1), � is the inhabitingregion, w�x� represents the density of a species, and we assume that �is fully surrounded by inhospitable areas, since the population density issubject to homogeneous Dirichlet boundary conditions. The operator �measures the diffusivity and the external transport effects of the species.In the case m > 1 the diffusion, i.e., the rate at which the species movesfrom high density regions to low density ones, is slower than in the linearcase (m = 1), which gives to rise a “more realistic” model. Moreover, hered > 0 is the diffusion rate of the species and b�x� and σ are associated

1 Research was partially financed by C.I.C.Y.T. Project MAR98-0486.

200

0022-247X/02 $35.00 2002 Elsevier Science (USA)All rights reserved.

logistic equation with nonlinear diffusion 201

Case p>1 Case p=1

λ λ1

|| || || ||

σ [L] σ [L+b]1

FIG. 1. Bifurcation diagrams with q = 1.

with the limiting effect of crowding on the population and the growth rateof the species, respectively.An appropriate change of variable, see (5), transforms (1) into

�u=λuq − b�x�up in �

u= 0 on ∂�(2)

with λ ∈ � 0 < q < p, and q ≤ 1. The case q = 1 and p ≥ 1 has beenwidely studied in recent years. When q = 1 and p > 1, it is well known thatthere exists a unique positive solution θλ of (2) if, and only if, λ > σ1�,where σ1� is the principal eigenvalue of � in � subject to homoge-neous Dirichlet boundary conditions. Moreover, there exists a continuumof positive solutions of (2) bifurcating from �λ u� = �σ1� 0�, which isunbounded. In the particular case q = p = 1 a vertical bifurcation diagramappears at λ = σ1� + b. Figure 1 shows these cases.When q < 1, to our knowledge only partial results are known about

the existence and uniqueness of positive solutions of (2). Indeed, when� = −� and b�x� = b ∈ �, it was proved in [12, Corollary 1] that thereexists a unique positive solution of (2) if, and only if, λ > 0. When b isa function in x and � = −�, Pozio and Tesei [16] showed that if λ > 0there exists a positive solution of (2). Moreover, if p ≥ 1 or p < 1 and λ islarge enough, then the positive solution is unique; see Theorem 5 of [16].Similar results were obtained by Leung and Fan in [10]; see Theorem 2.1there. We improve on these results in two ways: when � is a second-orderuniformly elliptic operator not necessarily self-adjoint and b is a functionin x, we prove that there exists a unique positive solution of (2) if, and onlyif, λ > 0. This solution will be denoted by θλ qp. Moreover, there exists acontinuum of positive solutions of (2) bifurcating from the trivial solutionu = 0 at λ = 0 which is unbounded; see Fig. 2.

202 delgado and suarez

|| ||

λ

FIG. 2. Bifurcation diagram with q < 1.

We can define the map

�q� � → C2 α0 �� q�λ� �= θλ qp

with �q�λ� = 0 if λ ≤ 0. We focus on the study of the map �q; specifically,we analyze the behavior of �q as λ ↓ 0+ and λ ↑ +∞, through the sin-gular perturbation theory. We generalize the results obtained when q = 1.Indeed, when q < 1 and q < p, we prove that if 1 < p,

�q�λ�λ1/�p−q�

→(

1b�x�

)1/�p−q�

uniformly on compact subsets of � as λ ↑ +∞ and

�q�λ� = O�λ1/�1−q�� as λ ↓ 0+�but if p < 1,

�q�λ� = O�λ1/�1−q�� as λ ↑ +∞and

�q�λ�λ1/�p−q�

→(

1b�x�

)1/�p−q�

uniformly on compact subsets of � as λ ↓ 0+; and if p = 1,

�q�λ� = λ1/�1−q��q�1��These results are a first step to obtaining nonexistence and existence resultsfor systems with nonlinear diffusion, as has already been shown in [4] whendiffusion is linear.Finally, we study how the bifurcation diagram of Fig. 2 varies when q ↑ 1.

We will show that if p > 1, θλ qp → θλ as q ↑ 1. In the special case p = 1,


we prove that if λ < σ1� + b (resp., λ > σ1� + b) then θλ qp tendsto 0 (resp. infinity) as q ↑ 1.An outline of this work is as follows. In Section 2 we study the existence

and uniqueness of a positive solution of (2), as well as some monotonyproperties of �q. In Section 3 we analyze the behavior of the mapping �qas λ ↓ 0+, λ ↑ +∞ (Theorem 3), and as q ↑ 1.

2. EXISTENCE AND COMPARISON RESULTS

In this section we study the positive solutions of

d�wm=σw − b�x�wr in �

w= 0 on ∂�(3)

where � is a bounded domain of �NN ≥ 1, with smooth boundary ∂�;m > 1; r > 1; d > 0; b ∈ Cα�� α ∈ �0 1�, with b�x� > 0 for all x ∈ ��; σis a real parameter; and � is a second-order operator of the form

� �= −N∑

i j=1aij

∂2

∂xi ∂xj+

N∑i=1bi∂

∂xi

with

aij ∈ C1 α�� bi ∈ Cα�� aij = aji with 0 < α < 1

and is uniformly elliptic in the sense that

∃ρ > 0 such thatN∑

i j=1aij�x�ξiξj ≥ ρ�ξ�2 ∀ ξ ∈ �N ∀x ∈ �� (4)

In the following, given any function f ∈ Cα��, we shall denote

fM �= sup��f fL �= inf

��f�

If r �= m, by performing the change

wm = dm/�r−m�u (5)

(3) can be rewritten as

�u=λuq − b�x�up in �

u= 0 on ∂�(6)

where p and q satisfy

�H� 0 < q < p q < 1�


In the special case r = m, the change wm = u transforms (3) into

�d� + b�x��u=λuq in �

u= 0 on ∂��(7)

On the other hand, it is well-known that the linear eigenvalue problem

�� + f �u=λu in �

u= 0 on ∂�(8)

with f ∈ L∞��, has a principal eigenvalue σ�1 � + f , with a correspond-ing eigenfunction ϕ�1 � + f �x� > 0 for all x ∈ �, and ∂nϕ�1 � + f �x� < 0for all x ∈ ∂� where n is the outward unit normal on ∂� and is normal-ized such that �ϕ�1 � + f �∞ = 1 (the superscript � will be omitted if noconfusion arises).The following results characterize the existence and uniqueness of posi-

tive solutions for (6) and (7).

Theorem 1. Assume �H�. Then (6) possesses a unique positive solutionin C2 α�� for some α ∈ �0 1� if, and only if, λ > 0.

Proof. We use the sub-supersolution method with Holder continuousfunctions; cf. [1] and Theorem 4.5.1 in [15]. It is not hard to show thatu �= �λ/bL�1/�p−q� is a supersolution of (6). Moreover, using the maximumprinciple we can prove that

�u�∞ ≤(λ

bL

) 1p−q

(9)

for any u solution of (6).Take u =� εϕ1�, with ε > 0 to be chosen. It is easy to check that we

can take ε > 0 sufficiently small such that u is a subsolution of (6) andu ≤ u. This proves the existence of a positive solution of (6) in C2 α�� forsome α ∈ �0 1�. The maximum principle implies that λ > 0 is a necessarycondition for the existence of a positive solution of (6). For the uniquenesspart of the proof we are going to use a change of variable already used in[17] (see also [3]) in a slightly different context. We define

z �= 11− qu

1−q�

Then (6) is equivalent to

�z− q

�1−q�zN∑

i j=1aij∂z

∂xi

∂z

∂xj=λ−b�x��1−q��p−q�/�1−q�

×z�p−q�/�1−q� in � (10)

z=0 on ∂��


Let z2 be the maximal solution of (10), which exists by (9). Suppose thereexists another solution z1 of (10) with z1 ≤ z2. We are going to prove thatz1 ≥ z2. We argue by contradiction.We suppose that there exists a P ∈ � where

' �= z1 − z2attains its negative minimum. Let r > 0 be such that 0 < z1�x� < z2�x� forall x ∈ B�P r�, where B�P r� is the ball of radius r centered at P . It is nothard to show that ' satisfies

�'− q

1− q( N∑i j=1

aij

[1z1

∂z1∂xi

∂z1∂xj

− 1z2

∂z2∂xi

∂z2∂xj

])

= −b�x��1− q��p−q�/�1−q�(z�p−q�/�1−q�1 − z�p−q�/�1−q�2

)�

On the other hand, it can be proved thatN∑

i j=1aij

[1z1

∂z1∂xi

∂z1∂xj

− 1z2

∂z2∂xi

∂z2∂xj

]=

N∑i=1ci∂'

∂xi− c�x�'

where

ci =N∑j=1aij

1z1

(∂z1∂xj

+ ∂z2∂xj

) c�x� = 1

z1z2

N∑i j=1

aij∂z2∂xi

∂z2∂xj

�

So, ' verifies

�1'+q

1− qc�x�' = −b�x��1− q��p−q�/�1−q�

(z�p−q�/�1−q�1

− z�p−q�/�1−q�2

) in B�P r� (11)

being

�1 = −N∑

i j=1aij

∂2

∂xi ∂xj+

N∑i=1

(bi −

q

1− qci) ∂

∂xi�

By (4), c�x� ≥ 0 in B�P r�, and from (H) we have that z�p−q�/�1−q�2 >

z�p−q�/�1−q�1 in B�P r�, and so by the strong maximum principle of Hopf(see for example Theorem 3.5 in [6]) ' = C < 0 in B�P r� with C con-stant. Thus, the left-hand side of (11) is nonpositive and the right one ispositive. This gives a contradiction and completes the proof.

The following result is well known when the operator is self-adjoint (see[2, 9, 10, and 17], for example), and its proof can be deduced by Theorem 1.So we only need to present an alternative uniqueness proof in which we usea singular eigenvalue problem.


Theorem 2. If 0 < q < 1, then (7) possesses a unique positive solution inC2α�� for some α ∈ �0 1� if, and only if, λ > 0.

Proof. Let u1 u2, u1 ≥ u2, u1 be the maximal positive solution of (7)and u2 be an arbitrary positive solution. Then

σ1[d� + b− λuq−1i

] = 0 i = 1 2� (12)

Observe that this principal eigenvalue is not in the setting of (8) becauseuq−1i �∈ L∞��. However, ui is a positive function satisfying (7) and so, by

the strong maximum principle, there exists a positive constant C such that

Cd��x� ≤ ui�x� for all x ∈ ��where d��x� �= dist�x ∂��. Hence, d1−q� �x�uq−1i is bounded and so we canapply the results of [8] (see also [5] for self-adjoint operators) to correctlydefine σ1d� + b− λuq−1i . Now, apply the mean value theorem

�d� + b− λqξq−1��u1 − u2� = 0

for some u2 ≤ ξ ≤ u1. Hence,

0 = σ1d� + b− λqξq−1 ≥ σ1d� + b− λquq−12 but from (12) we get that σ1d� + b− λquq−12 > σ1d� + b− λuq−12 = 0,which gives a contradiction.

In the following we denote by θλ qp the unique positive solution of (6)if (H) holds, with θλ qp = 0 if λ ≤ 0.

The following result is well known and it will be very useful for comparingpositive solutions of different logistic boundary value problems.

Lemma 1. Assume �H�. Then:1. If λ ≤ 0, (6) does not admit a positive subsolution.2. If λ > 0 and u is a positive supersolution of (6), then θλ qp ≤ u.3. If λ > 0 and u is a positive subsolution of (6), then u ≤ θλ qp.

From Lemma 1 we obtain the following results. The first one shows themonotony of θλ qp with respect to the domain and the second one willbe quite useful below.

Corollary 1. Assume (H) and let �1 be a subdomain of � with bound-ary ∂�1 sufficiently smooth. If we denote by θ

�λ qp the unique positive solu-

tion of (6) in �, then

θ�1λ qp < θ

�λ qp in �1�


Corollary 2. Assume �H�. Then there exists a constant K�λ� �= K��,λ qp� > 0 such that

K�λ�ϕ1� ≤ θλ qp <(λ

bL

) 1p−q� (13)

Proof. We will prove that Kϕ1� is a subsolution of (6). Then the firstinequality of (13) follows from Lemma 1. Indeed, Kϕ1� is a subsolutionof (6) if, for example,

K1−qσ1� + bMKp−q = λ� (14)

Now, for fixed λ > 0, (14) has a unique positive solution which we denoteK�λ� and which satisfies

limλ↓0+

K�λ� = 0 and limλ↑+∞

K�λ� = ∞�

The second inequality of (13) follows from (9) and the strong maximumprinciple.

Remark 1. It is important to note that:

1. If p = 1,

K�λ� =(

λ

σ1� + bM

) 11−q�

2. If 1 < p,

K�λ� = O�λ1/�1−q�� if λ ↓ 0+ and K�λ� = O�λ1/�p−q�� if λ ↑ +∞�3. If p < 1,

K�λ� = O�λ1/�p−q�� if λ ↓ 0+ and K�λ� = O�λ1/�1−q�� if λ ↑ +∞�When b�x� = b ∈ �, Lemma 1 can be used to prove some monotony

properties of θλ qp with respect to λ.

Proposition 1. Suppose �H� and that b�x� = b ∈ �, λµ > 0. Thefollowing assertions are true:

1. Assume 1 ≤ p. If λ ≥ µ, then(λ

µ

)1/�p−q�θµ qp ≤ θλ qp ≤

(λ

µ

)1/�1−q�θµ qp�

2. Assume p < 1. If λ ≥ µ, then(λ

µ

)1/�1−q�θµ qp ≤ θλ qp ≤

(λ

µ

)1/�p−q�θµ qp�


Proof. We only prove the first part; the second one follows similarly. So,assume 1 ≤ p and take η �= �λ/µ�1/�p−q�. It can be shown that ηθµ qp isa subsolution of (6). Analogously, it can be proved that �λ/µ�1/�1−q�θµ qpis a supersolution of (6). From Lemma 1, the result follows.

As an immediate consequence of Proposition 1, we obtain the followingresult.

Corollary 3. Assume �H� and that b�x� = b ∈ �. The following asser-tions are true:

1. θλ qp is increasing in λ.

2. If 1 < p, then

θλ qpλ1/�p−q�

is increasing in λ andθλ qpλ1/�1−q�

is decreasing in λ�

3. If p < 1, then


is decreasing in λ andθλ qpλ1/�1−q�

is increasing in λ�

4. If p = 1, then

θλ q 1λ1/�1−q�

is constant in λ�

Remark 2. 1. The case p = 1 is very special. In fact, it holds

θλ q 1 = λ1/�1−q�θ1 q 1� (15)

2. In the very special case q = 1 and p = 2, it was shown in [11] thatθλ 1 2/λ is increasing in λ. Thus, our result is a generalization of that one.

3. ASYMPTOTIC BEHAVIOR OF THE BRANCH θλ qp

We will regard (6) as a bifurcation problem with λ as the bifurcationparameter. By the above results, from the trivial state u = 0 emanates acurve of positive solutions at λ = 0. This curve goes to the right and toinfinity as λ ↑ +∞. Throughout this section ωλ q will denote the uniquepositive solution of (7) with d = 1 and b ≡ 0.The main result of this section completes the information of Corollary 3.


Theorem 3. Assume �H�.1. If 1 < p, then

limλ↓0+

θλ qpλ1/�1−q�

= ω1 q in C2��

limλ↑+∞


=(

1b�x�

)1/�p−q�uniformly on compacts of ��

2. If p < 1, then

limλ↓0+


=(

1b�x�

)1/�p−q�uniformly on compacts of ��

limλ↑+∞

θλ qpλ1/�1−q�

= ω1 q in C2��

3. If p = 1, then

limλ↓0+

θλ q 1λ1/�1−q�

= limλ↑+∞

θλ q 1λ1/�1−q�

= θ1 q 1�

To prove this result we need some preliminaries. Consider the problem

d�w=wq − b�x�wp in �

w= 0 on ∂�(16)

with d > 0. Observe that this problem is in the setting of (3) and so, for fixedd > 0, there exists a unique positive solution of (16) which we will denote'd qp. The following result provides us with the behavior of 'd qp asd ↑ +∞ and d ↓ 0+. This is a singular perturbation problem. In fact, wegive a proof that is a slight modification of Theorem 3.4 in [4]; we includeit for the reader’s convenience.

Theorem 4. Assume �H� and let 'd qp be the unique positive solutionof �16�. Then

limd↓0+

'd qp =(

1b�x�

) 1p−q

uniformly on compact subsets of �

limd↑+∞

'd qp = 0 uniformly on ��

(17)

Proof. We consider ud = d−1ω1 q. It is easy to show that ud is asupersolution of (16) provided that

ωq1 q

(1− d−q + b�x�d−pωp−q1 q

)≥ 0�

Taking d sufficiently large and a further application of Lemma 1 gives (17).


Let � be a compact subset of �. We shall show that given ε > 0 thereexists d0 = d0�Kε� > 0 such that for every d < d0,

(1b

) 1p−q− ε ≤ 'd qp ≤

(1b

) 1p−q+ ε in �� (18)

Let β = β�ε� be such that

0 < β�ε� <((

1b

)1/�p−q�+ ε

)p−q− 1b�

Take ' ∈ C∞�� such that

(1b+ β

) 1p−q≤ ' ≤

(1b

) 1p−q+ ε in ��

Then, we have

'q − b�x�'p = b�x�'q�1/b�x� −'p−q� ≤ −βb�x�'q ≤ d�' in �

for any d < d1, for some d1�ε�. Thus, for any d < d1 the function ' is asupersolution of (16), and from Lemma 1 we get

'd qp ≤ ' ≤(1b

) 1p−q+ ε�

By a compactness argument, to complete the proof of (18) it suffices toshow that given x0 ∈ � there exist r0 > 0 and d2 = d2�x0� such that foreach d < d2,

'd qp ≥(1b

) 1p−q− ε in B�x0 r0��

For any B�x0 r� ⊂ �, r > 0, from Corollary 1 we have

'B�x0 r�d qp ≤ 'd qp in B�x0 r��

Thus, to complete the proof it remains to show that for any d < d2,

'B�x0 2r0�d qp ≥

(1b

) 1p−q− ε in B�x0 r0��


We consider two different cases.

Case 1. Suppose there exists r0 > 0 such that b�x� = b ∈ � in B0 �=B�x0 2r0� ⊂ �. Let ϕB0

1 � be normalized so that∥∥∥ϕB01 �

∥∥∥∞ B0

= 12� (19)

Set B1 �= B�x0 r0�. Then, ϕB01 ��x� > 0 for each x ∈ �B1 and there exists

a ϕ0 ∈ C2�B1� such that

ϕ0�x0� = 1 �ϕ0�∞ B1= 1 ϕ0�x� > 0 ∀x ∈ �B1 (20)

and the function /� B0 → � defined by

/�x� =ϕB01 ��x� if x ∈ B0\B1

ϕ0�x� if x ∈ �B1

lies in C2�B0�. Given δ ∈ �0 1�, we define

/δ �= δ(1b

) 1p−q/�

Since b ∈ �, then /δ ∈ C2�B0�. It is not hard to show that /δ is a positivesubsolution of (16) if, and only if,

�/

/q ≤1db�1−q�/�p−q�δq−1�1− δp−q/p−q� in B0 (21)

and this inequality holds if d is sufficiently small. Indeed, observe that theleft-hand side of (21) is bounded above in B0. From (19) and (20), we havethat / ≤ /q, and so

�/

/q ≤�/

/≤ C

for some C > 0. This last inequality follows by the strong maximum prin-ciple. Thus, since δ < 1 and 0 ≤ / ≤ 1, it is sufficient to take d small tosatisfy (21). From Lemma 1, we have that for d sufficiently small

/δ ≤ 'B0d qp ≤ 'd qp in B0�

Clearly, since /�x0� = 1 if δ is taken sufficiently close to 1, then /δ willbe as close as we want to �1/b�1/�p−q� on some ball centered at x0. Thiscompletes the proof in this case.


Case 2� Assume b�x� is not constant in some ball centered at x0. Wehave

d�'B0dqp =

('B0dqp

)q−b�x�

('B0dqp

)p≥('B0dqp

)q−bMB0

('B0dqp

)p

and so 'B0d qp is a positive supersolution of (16) with b�x� = bMB0

∈ �,and so from Lemma 1 we have that

'B0d qp ≥ 'B0

d qp

where 'B0d qp stands for the unique positive solution of (16) with b�x� =

bMB0∈ �. Thus, from Case 1, there exists r1 > 0 such that

'B0d qp ≥ 'B0

d qp ≥ �1/bMB0�1/�p−q� − ε

2in B�x0 r1��

Therefore, if B0 is chosen such that for each x ∈ B0,

�1/bMB0�1/�p−q� ≥ �1/b�x��1/�p−q� − ε

2

then

'B0d qp ≥

(1b�x�

)1/�p−q�− ε

for each x ∈ B�x0 r1�. This completes the proof.

We consider the equation

�w=wq − db�x�wp in �

w= 0 on ∂��(22)

From Theorem 1, given d > 0 there exists a unique positive solution 1d qpof (22). The following result provides us with the behavior of 1d qp asd ↓ 0+ and d ↑ +∞.

Theorem 5. Assume �H� and let 1d qp be the unique positive solutionof �22�. Then,

limd↓0+

1d qp = ω1 q in C2 ν�� for some ν ∈ �0 1�

limd↑+∞

1d qp = 0 uniformly on ��


Proof. By Corollary 2,

1d qp ≤(

1dbL

)1/�p−q�

from which the second relation follows.On the other hand, it is not hard to prove that u = ω1 q is a supersolu-

tion of (22) and, hence,

�1d qp�∞ ≤ �ω1 q�∞ = K (independent of d)�

Thus, according to the Ls theory of elliptic equations, �1d qp�d is abounded sequence in W 2 s��, for s > 1, and so we can extract a con-vergent subsequence, again labeled by d, such that

1d qp → w in C1 α�� where 0 < α = 1−N/s < 1

as d ↓ 0+. Using (22) we get

1d qp = ��−1(1qd qp − db�x�1pd qp

)

and so

�w = wq in �

w = 0 on ∂��

Now, as in Corollary 2, we can get a constant K = K�� > 0, indepen-dent of d, such that

K��ϕ1� ≤ 1d qp for all d ∈ 0 d0 for some d0 > 0�

In fact, in this case we can take K satisfying

dbMKp−q +K1−qσ1� = 1�

It can be proved that the map

d ∈ 0 d0 → K�d�is continuous, and so there exists the constant K��. We can deducethat w = ω1 q, and by the Ascoli–Arzela Theorem all sequences convergein C2 ν�� for some ν ∈ �0 1�, and the result follows.

Proof of Theorem 3� Let us define

/λ qp �=θλ qpλ1/�p−q�

�


It is easy to check that /λ qp is the unique positive solution of the equa-tion

1λ�p−1�/�p−q�

�w=wq − b�x�wp in �

w= 0 on ∂�

included in the setting (16). Now, Theorem 4 proves two relations ofTheorem 3. If we write

χλ qp �=θλ qpλ1/�1−q�

then χλ qp is the unique positive solution of

�w=wq − λ�p−1�/�1−q�b�x�wp in �

w= 0 on ∂��

From Theorem 5, the other relations follow.Finally, for p = 1 the result follows by (15). The proof of Theorem 3 is

completed.

Now, we denote by θλ the unique positive solution of (6) for q = 1 andp > 1 if λ > σ1�, with θλ = 0 if λ ≤ σ1�. The next results provideus with the behavior of θλ qp as q ↑ 1. We consider two different cases,p > 1 and p = 1.

Theorem 6. Assume p > 1 > q and λ > 0. Then

limq↑1

θλ qp = θλ in C2 ν�� for some ν ∈ �0 1��

Proof. Fix δ ∈ �0 1�. We know from Corollary 2 that for q ∈ 1− δ 1,

�θλ qp�∞ ≤(λ

bL

) 1p−q≤ K (independent of q).

We can reason as in Theorem 5 and conclude that there exists a subse-quence �θλ qp�q such that

θλ qp → w ≥ 0 in C1 α�� with 0 < α < 1

as q ↑ 1, with w satisfying

�w=λw − b�x�wp in �

w= 0 on ∂��

So, if λ ≤ σ1�, w = 0. On the other hand, if λ > σ1�, we can chooseK�λ�, independent of q, such that

K�λ�ϕ1� ≤ θλ qp�Again the Ascoli–Arzela Theorem completes the proof.


The case p = 1 is more complicated. We are going to prove that θλ qptends to 0 when λ < σ1� + b and to infinity when λ > σ1� + b as q ↑ 1,showing that the bifurcation diagram with q < 1 (see Fig. 2) “converges”to the one with q = p = 1 (see Fig. 1).

Theorem 7. Assume 0 < q < p = 1. Then:

1. If λ < σ1� + b, then �θλ q 1�∞ → 0 as q ↑ 1.2. If λ > σ1� + b, then �θλ q 1�∞ → ∞ as q ↑ 1.

Proof. For the first part, we fix λ < σ1� + b. From the continuousdependence of σ1� + b with respect to the domain, there exists a regulardomain �′ ⊃ � such that

λ < σ�′

1 � + b < σ�1 � + b� (23)

Let ϕ′1 �= ϕ�′

1 � + b with �ϕ′1�∞�′ = 1. It is not difficult to see thatu �=Mϕ′1 is a supersolution of (6) is

M =(

λ

σ�′

1 � + b

)1/�1−q� 1�ϕ′1�L�

and so, by Lemma 1,

�θλ q 1�∞� ≤M�ϕ′1�∞��Now, it suffices to use (23) and to tend q ↑ 1.

For the second part, we are going to build a subsolution whose normtends to infinity. We take ϕ1� + b normalized such that �ϕ1� +b�∞ = 1. It is easy to prove that u �= Cϕ1� + b is a subsolution of (6)with

C =(

λ

σ1� + b)1/�1−q�

�

Again, taking q ↑ 1, the proof concludes since λ > σ1� + b.

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on the structure of the positive solutions of the logistic equation with nonlinear diffusion

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