ground state sign-changing solutions for kirchhoff equations with...

Electronic Journal of Qualitative Theory of Differential Equations2019, No. 47, 1–13; https://doi.org/10.14232/ejqtde.2019.1.47 www.math.u-szeged.hu/ejqtde/

Ground state sign-changing solutions forKirchhoff equations with logarithmic nonlinearity

Lixi Wen, Xianhua Tang and Sitong ChenB

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R. China

Received 24 December 2018, appeared 26 July 2019

Communicated by Patrizia Pucci

Abstract. In this paper, we study Kirchhoff equations with logarithmic nonlinearity:{−(a + b

∫Ω |∇u|

2)∆u + V(x)u = |u|p−2u ln u2, in Ω,u = 0, on ∂Ω,

where a, b > 0 are constants, 4 < p < 2∗, Ω is a smooth bounded domain of R3 andV : Ω → R. Using constraint variational method, topological degree theory and somenew energy estimate inequalities, we prove the existence of ground state solutions andground state sign-changing solutions with precisely two nodal domains. In particular,some new tricks are used to overcome the difficulties that |u|p−2u ln u2 is sign-changingand satisfies neither the monotonicity condition nor the Ambrosetti–Rabinowitz condi-tion.

Keywords: logarithmic nonlinearity, ground state solution, sign-changing solution,Kirchhoff equations.

2010 Mathematics Subject Classification: 35J20, 35J65.

1 Introduction

In this paper, we investigate the following Kirchhoff equation with logarithmic nonlinearity:{−(a + b

∫Ω |∇u|

2)∆u + V(x)u = |u|p−2u ln u2, in Ω,u = 0, on ∂Ω,

(1.1)

where a, b > 0 are constants, 4 < p < 2∗, Ω is a smooth bounded domain of R3 and V : Ω→ Rsatisfies

(V) V ∈ C(Ω, R) and infx∈Ω V(x) > 0.

In the past years, there have been increasing interests in studying logarithmic nonlinearity dueto its relevance in quantum mechanics, quantum optics, nuclear physics, transport and diffu-sion phenomena, open quantum systems, effective quantum gravity, theory of superfluidityand Bose–Instein consideration (see [32] and the references therein).BCorresponding author. Email: [email protected]

https://doi.org/10.14232/ejqtde.2019.1.47https://www.math.u-szeged.hu/ejqtde/

2 L. Wen, X. Tang and S. Chen

Denote by H10(Ω) the Sobolev space equipped with the norm and inner product

‖u‖ =(∫

Ωa|∇u|2 + V(x)u2dx

) 12

, 〈u, v〉 =∫

Ω[a∇u · ∇v + V(x)uv]dx,

under the assumption (V). This norm is equivalent to the standard norm of H10(Ω).Define the energy functional I : H10(Ω)→ R

I(u) =12

∫Ω

[a|∇u|2 + V(x)u2

]dx +

b4

(∫Ω|∇u|2dx

)2+

2p2

∫Ω|u|pdx− 1

p

∫Ω|u|p ln u2dx.

(1.2)

By elementary computation, we have

limt→0

tp−1 ln t2

t= 0 and lim

t→∞

tp−1 ln t2

tq−1= 0, (1.3)

where q ∈ (p, 2∗). Then for any ε > 0, there exists Cε > 0 such that

|t|p−1| ln t2| ≤ ε|t|+ Cε|t|q−1, ∀t ∈ R\{0}. (1.4)

By a similar argument of [23] and (1.4), we have that I ∈ C1(H10(Ω), R) and

〈I′(u), v〉 =∫

Ω[a|∇u · ∇v + V(x)uv] dx + b

∫Ω|∇u|2dx

∫Ω∇u · ∇vdx

−∫

Ω|u|p−2uv ln u2dx

(1.5)

for all u, v ∈ H10(Ω). u ∈ H10(Ω) is a weak solution of (1.1) if and only if u is a critical point ofI. Moreover, if u ∈ H10(Ω) is a solution of (1.1) and u± 6= 0, then u is a sign-changing solutionof (1.1), where

u+(x) := max{u(x), 0}, u−(x) := min{u(x), 0}.

From (1.5), one has

〈I′(u), u±〉 =∫

Ω

[a|∇u±|2 + V(x)(u±)2

]dx + b

∫Ω|∇u|2dx

∫Ω|∇u±|2dx

−∫

Ω|u±|p ln(u±)2dx.

(1.6)

As we know, (1.1) is a special form of the following Kirchhoff type problem:{−(a + b

∫Ω |∇u|

2)∆u + V(x)u = f (u), in Ω,

u = 0, on Ω,(1.7)

where f ∈ C(R, R). System (1.7) is related to the stationary analogue of the Kirchhoff equation

utt −(

a + b∫

Ω|∇u|2dx

)∆u = f (x, u) (1.8)

proposed by Kirchhoff in [14] as an extension of the classical D’Alembert’s wave equationsfor free vibration of elastic strings. For more mathematical and physical background of theproblem (1.7), we refer the readers to the papers [1, 2, 4, 5] and the references therein.

Ground state sign-changing solutions for Kirchhoff equations with logarithmic nonlinearity 3

Kirchhoff equation (1.8) received increasingly more attention after Lion’s [15] proposed anabstract functional analysis framework to it. There are many important results on the existenceof positive solutions, multiple solutions and ground state solutions for Kirchhoff equations,see for example, [6–9, 11, 12, 18, 19, 25–27, 29, 30] and the references therein.

Recently, many researchers began to study the sign-changing solutions for (1.7). WhenV(x) ≡ 0, Zhang [31] obtained sign-changing solutions for (1.7) via invariant sets of descentflow under the following (AR) condition

(AR) there exists υ > 4 such that υF(x, t) ≤ t f (x, t) for |t| large, where F(x, t) =∫ t

0 f (x, s)ds.

Shuai [22] proved the existence of sign-changing solutions for system (1.7) when f (x, u) =f (u) satisfies the Nehari type monotonicity condition:

(F) f (t)|t|3 is increasing on (−∞, 0) ∪ (0,+∞)

and some other conditions. To obtain a constant sign solution and a sign-changing solutionfor the following Kirchhoff-type equation:{

−M(∫

Ω |∇u|2dx)

∆u = λ f (u), in Ω,

u = 0, on Ω,(1.9)

Lu [17] also proposed the following monotonicity condition

(F’) there exists µ ∈ (2, 2∗) such that f (t)|t|µ−2t is nondecreasing in |t| > 0.

In particular, letting a = 1 and b = 0 in (1.1) leads to the following Schrödinger equation:{−∆u + V(x)u = |u|p−2u ln u2, x ∈ Ω,u ∈ H10(Ω).

(1.10)

System (1.10) has received much attention in mathematical analysis and applications. D’Avenia[3] proved the existence of infinitely many solutions of (1.10) with p = 2 in the framework ofthe non-smooth critical point theory, which is developed by Degiovanni and Zani [10]. Whenp = 2 and V satisfies the following condition

(V’) V ∈ C(RN , R), lim|x|→∞ V(x) = supx∈RN V(x) := V∞ ∈ (−1, ∞) and the spectrumσ(−∆ + V + 1) ⊂ (0, ∞).

Ji [13] obtained a positive ground state solution of (1.10). For more results on the logarithmicSchrödinger equation, we refer the readers to [23, 24] and the references therein.

Motivated by the works mentioned above, in the present paper, we intend to prove theexistence of ground state solutions and sign-changing solutions for (1.1). It is worth pointingout that the methods used in [17, 22, 31] rely heavily on the monotonicity conditions (F), (F’)or (AR) condition, so their methods do not work for (1.1) because f (x, u) = |u|p−2u ln u2satisfies neither the monotonicity conditions (F), (F’) or (AR) condition. Furthermore, due tothe existence of the nonlocal term

(∫Ω |∇u|

2dx)

∆u, the method dealing with (1.10) can not beapplicable for (1.1). Therefore, a natural question is whether we can still find sign-changingsolutions for Kirchhoff equation with logarithmic nonlinearity. The present paper will give anaffirmative response and establish the relation between the energy of sign-changing solutionsand ground state solutions of (1.1). To the best of our knowledge, there are only a few resultsof sign-changing solutions for system (1.1).

Now, we state the main result.


Theorem 1.1. Assume that (V) holds. Then problem (1.1) has a sign-changing solution ũ ∈ M withprecisely two nodal domains such that I(ũ) = infM I := m, where

M = {u ∈ H10(Ω), u± 6= 0, and 〈I′(u), u+〉 = 〈I′(u), u−〉 = 0}.

Theorem 1.2. Assume that (V) holds. Then problem (1.1) has a ground state solution ū ∈ N suchthat I(ū) = infN I := c, where

N = {u ∈ H10(Ω)\{0}, 〈I′(u), u〉 = 0}.

Moreover, m ≥ 2c.

To obtain this result, we must overcome the following difficulties:

1) The fact that |u|p−2u ln u2

u3 is not increasing prevent us from using the Nehari manifold methodin [17, 22].

2) It is more complicated to show the boundedness of minimizing sequences of c = infN Iand m = infM I.

3) Compared with the case that a = 1 and b = 0, the presence of the nonlocal term(∫Ω |∇u|

2dx)

∆u brings us some new troubles. More specifically, the functional:

χ : H10(Ω)→ R : u 7→∫

Ω|∇u|2dx

∫Ω∇u · ∇v

is not weakly continuous for any v ∈ H10(Ω), which cause great obstacles when provingthat the limit of (PS)c sequence is indeed a nontrivial solution of (1.1).

Next, we give some notations. We denote the ball centered at x with the radius r by B(x, r)and the norm of Li(Ω) is denoted by | · |i for 1 ≤ i < ∞. We shall denote by Ci, i = 1, 2, . . . forvarious positive constants.

2 Preliminary lemmas

Firstly, we establish an energy estimate inequality related to I(u), I(su+ + tu−), 〈I′(u), u+〉and 〈I′(u), u−〉 to overcome the difficulty that the logarithmic nonlinearity |u|p−2u ln u2 doesnot satisfy (F).Lemma 2.1. For all u ∈ H10(Ω) and s, t ≥ 0, there holds

I(u) ≥ I(su++tu−)+ 1−sp

p〈I′(u), u+〉+ 1−t

p

p〈I′(u), u−〉+

(1−s2

2− 1−s

p

p

)‖u+‖2

+

(1−t2

2− 1−t

p

p

)‖u−‖2 + b

[(1− s4

4− 1− s

p

p

)|∇u+|42 +

(1− t4

4− 1− t

p

p

)|∇u−|42

]+ b

sp + tp − 2s2t24

|∇u+|22|∇u−|22. (2.1)

Proof. It is obvious that (2.1) holds for u = 0, then we consider the case u 6= 0. Through apreliminary calculation, we have

2(1− τp) + pτp ln τ2 > 0, ∀ τ ∈ (0, 1) ∪ (1,+∞). (2.2)


SetΩ+u = {x ∈ Ω : u(x) ≥ 0}, Ω−u = {x ∈ Ω : u(x) < 0}.

For u ∈ H10(Ω)\{0}, one has∫Ω|su+ + tu−|p ln(su+ + tu−)2dx

=∫

Ω+|su+ + tu−|p ln(su+ + tu−)2dx +

∫Ω−|su+ + tu−|p ln(su+ + tu−)2dx

=∫

Ω+|su+|p ln(su+)2dx +

∫Ω−|tu−|p ln(tu−)2dx

=∫

Ω

[|su+|p ln(su+)2 + |tu−|p ln(tu−)2

]dx

=∫

Ω

[|su+|p

(ln(u+)2 + ln s2

)+ |tu−|p

(ln(u−)2 + ln t2

)]dx, ∀ s, t ≥ 0. (2.3)

It follows from (1.2), (1.6), (2.2) and (2.3) that

I(u)− I(su+ + tu−)

=12

(‖u‖2−‖su+ + tu−‖2

)+

b4(|∇u|42 − |∇(su+ + tu−)|42) +

2p2

∫Ω

[|u|p−|su+ + tu−|p

]dx

− 1p

∫Ω

[|u|p ln u2−|su+ + tu−|p ln(su+ + tu−)2

]dx

=1− s2

2‖u+‖2 + 1− t

2

2‖u−‖2 + b(1− s

4)

4|∇u+|42 +

b(1− t4)4

|∇u−|42 +b(1− s2t2)

2|∇u+|22|∇u−|22

+2p2

∫Ω

[|u+|p − |su+|p + |u−|p − |tu−|p

]dx

− 1p

∫Ω

[|u+|p ln(u+)2 − |su+|p ln(u+)2 − |su+|p ln s2

]dx

− 1p

∫Ω

[|u−|p ln(u−)2 − |tu−|p ln(u−)2 − |tu−|p ln t2

]dx

=1− sp

p〈I′(u), u+〉+ 1− t

p

p〈I′(u), u−〉+

(1− s2

2− 1− s

p

p

)‖u+‖2 +

(1− t2

2− 1− t

p

p

)‖u−‖2

+ b[(

1− s44− 1− s

p

p

)|∇u+|42 +

(1− t4

4− 1− t

p

p

)|∇u−|42 +

sp + tp − 2s2t24

|∇u+|22|∇u−|22]

+2(1− sp) + psp ln s2

p2

∫Ω|u+|pdx + 2(1− t

p) + ptp ln t2

p2

∫Ω|u−|pdx

≥ 1− sp

p〈I′(u), u+〉+ 1− t

p

p〈I′(u), u−〉+

(1− s2

2− 1− s

p

p

)‖u+‖2 +

(1− t2

2− 1− t

p

p

)‖u−‖2

+ b[(

1− s44− 1− s

p

p

)|∇u+|42 +

(1− t4

4− 1− t

p

p

)|∇u−|42 +

sp + tp − 2s2t24

|∇u+|22|∇u−|22]

.

(2.4)

Hence, (2.1) holds for all u ∈ H10(Ω) and s, t ≥ 0.

Let s = t in (2.1), we can obtain the following corollary.

Corollary 2.2. For all u ∈ H10(Ω) and t ≥ 0, there holds

I(u) ≥ I(tu) + 1− tp

p〈I′(u), u〉+

(1− t2

2− 1− t

p

p

)‖u‖2 + b

(1− t4

4− 1− t

p

p

)|∇u|42. (2.5)


In view of Lemma 2.1 and Corollary 2.2, we have the following corollaries.

Corollary 2.3. For any u ∈ M, there holds I(u) = maxs,t≥0 I(su+ + tu−).

Corollary 2.4. For any u ∈ N , there holds I(u) = maxt≥0 I(tu).

Lemma 2.5. For any u ∈ H10(Ω)\{0}, there exists an unique tu > 0 such that tuu ∈ N .

Proof. First, we prove the existence of tu. Let u ∈ N be fixed and define a function g(t) =〈I′(tu), tu〉 on [0,+∞). Then,

g(t) = 〈I′(tu), tu〉 = t2‖u‖2 + bt4|∇u|42 −∫

Ω|tu|p ln(tu)2, ∀ t > 0. (2.6)

It follows from (1.4) and (2.6) that limt→0+ g(t) = 0, g(t) > 0 for t > 0 small and g(t) < 0 for tlarge. Since g(t) is continuous, there exits tu > 0 such that g(tu) = 0.

Next, we prove the uniqueness of tu. Arguing by contradiction, we suppose that thereexists u ∈ H10(Ω)\{0} and two positive constants t1 6= t2 such that g(t1) = g(t2). Sincefunction f (x) = 1−a

x

x is monotonically decreasing on (0,+∞) for a > 0 and a 6= 1, by (2.5),one has

I(t1u) ≥ I(t2u) +tp1 − t

p2

tp1〈I′(t1u), t1u〉+ t21

[1− ( t2t1 )

2

2−

1− ( t2t1 )p

p

]‖u‖2

+ bt4[

1− ( t2t1 )4

4−

1− ( t2t1 )p

p

]|∇u|42

> I(t2u),

and

I(t2u) ≥ I(t1u) +tp2 − t

p1

tp2〈I′(t2u), t2u〉+ t22

[1− ( t1t2 )

2

2−

1− ( t1t2 )p

p

]‖u‖2

+ bt4[

1− ( t1t2 )4

4−

1− ( t1t2 )p

p

]|∇u|42

> I(t1u).

This contradiction shows that tu > 0 is unique for any u ∈ H10\{0}.

Lemma 2.6. For any u ∈ H10(Ω) with u± 6= 0, there exists an unique pair (su, tu) of positive numberssuch that suu+ + tuu− ∈ M.

Proof. For any u ∈ H10(Ω) with u± 6= 0, Let

g1(s, t) = s2‖u+‖2 + bs4|∇u+|42 + bs2t2|∇u+|22|∇u−|22 −∫

Ω|su+|p ln(su+)2dx, (2.7)

andg2(s, t) = t2‖u−‖2 + bt4|∇u−|42 + bs2t2|∇u+|22|∇u−|22 −

∫Ω|tu−|p ln(tu−)2dx. (2.8)

Using (1.4), it’s easy to verify that g1(s, s) > 0 and g2(s, s) > 0 for s > 0 small and g2(t, t) < 0and g2(t, t) < 0 for t > 0 large enough. Thus, there exist 0 < r < R such that

g1(r, r) > 0, g2(r, r) > 0; g1(R, R) < 0, g2(R, R) < 0. (2.9)


From (2.7), (2.8), (2.9), we have

g1(r, t) > 0, g1(R, t) < 0 ∀ t ∈ [r, R]; (2.10)

andg2(s, r) > 0, g2(s, R) < 0 ∀ s ∈ [r, R]. (2.11)

In view of Miranda’s Theorem [20], there exists some point (su, tu) with r < su, tu < R suchthat g1(su, tu) = g2(su, tu) = 0, which implies suu+ + tuu− ∈ M. Using (2.1), as a similarargument of Lemma 2.5, we can obtain the uniqueness of (su, tu).

From Corollaries 2.3, 2.4, Lemmas 2.5 and 2.6, we can obtain the following lemma.

Lemma 2.7. The following minimax characterizations hold

infu∈N

I(u) =: c = infu∈H10 (Ω),u 6=0

maxt≥0

I(tu);

andinf

u∈MI(u) =: m = inf

u∈H10 (Ω),u± 6=0maxs,t≥0

I(su+ + tu−).

Lemma 2.8. c > 0 and m > 0 are achieved.

Proof. For every u ∈ N , we have 〈I′(u), u〉 = 0. Then by (1.4), (1.5) and the Sobolev embeddingtheorem, we get

‖u‖2 ≤ ‖u‖2 + b|∇u|42 =∫

Ω|u|p ln u2dx ≤ 1

2‖u‖2 + C1‖u‖q, (2.12)

which implies that there exists a constant α > 0 such that ‖u‖ ≥ α.Let {un} ⊂ M be such that I(un)→ m. By (1.2) and (1.5), one has

m + o(1) = I(un)−1p〈I′(un), un〉

=

(12− 1

p

)‖un‖2+b

(14− 1

p

)|∇un|42 +

2p2

∫Ω|un|pdx ≥

(12− 1

p

)‖un‖2. (2.13)

This shows that {‖un‖} is bounded. Thus, passing to a subsequence, we may assume thatu±n ⇀ ũ± in H10(Ω) and u

±n → ũ± in Ls(Ω) for 2 ≤ s < 2∗. Since {un} ⊂ M, we have

〈I′(un), u±n 〉 = 0. Similar as (2.12), there exists a constant β > 0 such that ‖u±n ‖ ≥ β. Using(1.4), (1.6) and the boundedness of {un}, we have

β2 ≤ ‖u±n ‖2 ≤ ‖u±n ‖2 + b|∇un|22|∇u±n |22 =∫

Ω|u±n |p ln(u±n )2dx ≤

β2

2+ C2

∫Ω|u±n |qdx.

Thus, ∫Ω|u±n |qdx ≥

β2

2C2.

By the compactness of the embedding H10(Ω) ↪→ Ls(Ω) for 2 ≤ s < 2∗, we get∫Ω|ũ±|qdx ≥ β

2

2C2,


which implies ũ± 6= 0. By (1.4), (1.5), [28, A.2], the Lebesgue dominated convergence theoremand the weak semicontinuity of norm, we have

‖u±‖2 + b|∇u|22|∇u±|22 ≤ limn→∞

(‖u±n ‖2 + b|∇un|22|∇u±n |2

)= lim

n→∞

∫Ω|u±n |p ln(u±n )2dx (2.14)

=∫

Ω|u±|p ln(u±)2dx, (2.15)

which, together with (1.6), implies

〈I′(ũ), ũ+〉 ≤ 0 and 〈I′(ũ), ũ−〉 ≤ 0. (2.16)

In view of Lemma 2.6, there exist two constants s̃, t̃ > 0 such that

s̃ũ+ + t̃ũ− ∈ M and I(s̃ũ+ + t̃ũ−) ≥ m. (2.17)

Thus, it follows from (1.2), (1.5), (2.1), (2.16), (2.17) and the weak semicontinuity of norm that

m = limn→∞

[I(un)−

1p〈I′(un), un〉

]= lim

n→∞

[(12− 1

p

)‖un‖2 + b

(14− 1

p

)|∇un|42 +

2p2

∫Ω|un|pdx

]≥(

12− 1

p

)‖ũ‖2 + b

(14− 1

p

)|∇ũ|42 +

2p2

∫Ω|ũ|pdx

= I(ũ)− 1p〈I′(ũ), ũ〉

≥ I(s̃ũ+ + t̃ũ−) + 1− s̃p

p〈I′(ũ), ũ+〉+ 1− t̃

p

p〈I′(ũ), ũ−〉 − 1

p〈I′(ũ), ũ〉

≥ m− s̃p

p〈I′(ũ), ũ+〉 − t̃

p

p〈I′(ũ), ũ−〉 ≥ m.

This shows〈I′(ũ), ũ±〉 = 0, I(ũ) = m,

i.e. ũ ∈ M and I(ũ) = m. Since ũ± 6= 0, then it follows from (2.1) that

m = I(ũ) ≥ 1p〈Φ′(ũ), ũ+〉+ 1

p〈Φ′(ũ), ũ−〉+

(12− 1

p

)‖ũ+‖2 +

(12− 1

p

)‖ũ−‖2 > 0.

By a similar argument as above, we have that c > 0 is achieved.

Lemma 2.9. The minimizers of infN I and infM I are critical points of I.

Proof. Assume that ū = ū+ + ū− ∈ M, I(ū) = m and I′(ū) 6= 0. Then there exist δ > 0 and$ > 0 such that

‖I′(u)‖ ≥ $, for all ‖u− ū‖ ≤ 3δ and u ∈ H10(Ω).

Let D = (1/2, 3/2)× (1/2, 3/2). By Lemma 2.1, one has

χ := max(s,t)∈∂D

I(sū+ + tū−) < m. (2.18)

For ε := min{(m− χ)/3, $δ/8} and S := B(ū, δ), [28, Lemma 2.3] yields a deformation η ∈C([0, 1]× H10(Ω), H10(Ω)) such that


(i) η(1, u) = u i f I(u) < m− 2ε or I(u) > m + 2ε;

(ii) η(1, Im+ε ∩ B(ū, δ)) ⊂ Im−ε;

(iii) I(η(1, u)) ≤ I(u), ∀ u ∈ H10(Ω).

By Lemma 2.1 and (iii), we have

I(η(1, sū+ + tū−)) ≤ I(sū+ + tū−) < I(ū)= m, ∀ s, t > 0, |s− 1|2 + |t− 1|2 ≥ δ2/‖ū‖2. (2.19)

By Corollary 2.3, we can obtain that I(sū+ + tū−) ≤ I(ū) = m for s, t > 0, then it follows from(ii) that

I(η(1, sū+ + tū−)) ≤ m− ε, ∀ s, t ≥ 0, |s− 1|2 + |t− 1|2 < δ2/‖ū‖2. (2.20)

Thus, it follows from (2.19) and (2.20) that

max(s,t)∈D

I(η(1, sū+ + tū−)) < m. (2.21)

Define h(s, t) = sū+ + tū−. We now prove that η(1, h(D)) ∩M 6= ∅, contradicting to thedefinition of m. Let β(s, t) := η(1, h(s, t)),

Ψ1(s, t) :=(〈I′(h(s, t)), ū+〉, 〈I′(h(s, t)), ū−〉

)and

Ψ2(s, t) :=(

1s〈I′(β(s, t)), (β(s, t))+〉, 1

t〈I′(β(s, t)), (β(s, t))−〉

).

Since ū ∈ M, by Lemma (2.6), (s, t) = (1, 1) is the unique pair of positive numbers such thatsū+ + tū− ∈ M. Furthermore, that sū+ + tū− ∈ M is equivalent to that (s,t) is a solution ofthe following equation

Ψ1(s, t) = (0, 0). (2.22)

Therefore, (2.22) has an unique solution (s, t) = (1, 1) in D. By virtue of the degree theory,we can derive that deg(Ψ1, D, (0, 0)) = 1. It follows from (2.18) and (i) that β = h on ∂D.Consequently, we get

deg(Ψ2, D, (0, 0)) = deg(Ψ1, D, (0, 0)) = 1,

which implies that Ψ2(s0, t0) = 0 for some (s0, t0) ∈ D, that is η(1, h(s0, t0)) = β(s0, t0) ∈ M.This contradiction shows that I′(ū) = 0.

In a similar way as above, we can prove that any minimizer of infN I is a critical pointof I.

3 Proof of Theorem 1.1

In view of Lemmas 2.8 and 2.9, there exist ũ ∈ M such that

I(ũ) = m, I′(ũ) = 0. (3.1)

Now, we show that ũ has exactly two nodal domains. Set ũ = u1 + u2 + u3, where

u1 ≥ 0, u2 ≤ 0, Ω1 ∩Ω2 = ∅, u1|RN\Ω1 = u2|RN\Ω2 = u3|Ω1∪Ω2 = 0, (3.2)


Ω1 := {x ∈ Ω : u1(x) > 0}, Ω2 := {x ∈ Ω : u2(x) < 0},and Ω1, Ω2 are connected open subset of Ω. Setting v = u1 + u2, we have that v+ = u1 andv− = u2, i.e. v± 6= 0. Note that I′(ũ) = 0, by a preliminary calculation, we can obtain

〈I′(ũ), v+〉 = −b|∇v+|22|∇u3|22, (3.3)

and〈I′(ũ), v−〉 = −b|∇v−|22|∇u3|22. (3.4)

It follows from (1.2), (1.5), (2.1), (3.1), (3.2), (3.3) and (3.4) that

m = I(ũ) = I(ũ)− 1p〈I′(ũ), ũ〉

= I(v) + I(u3) +b2|∇u3|22|∇u|22 −

1p[〈I′(v), v〉+ 〈I′(u3), u3〉+ 2b|∇u3|22|∇v|22

]≥ sup

s,t≥0

[I(sv+ + tv−) +

1− spp〈I′(v), v+〉+ 1− t

p

p〈I′(v), v−〉

]+ I(u3)−

1p〈I′(v), v〉 − 1

p〈I′(u3), u3〉

≥ sups,t≥0

[I(sv+ + tv−) +

bsp

p|∇v+|22|∇u3|22 +

btp

p|∇v−|22|∇u3|32

]+ (

12− 1

p)‖u3‖2 + b

(14− 1

p

)|∇u3|42 +

2p2

∫Ω|u3|pdx

≥ maxs,t≥0

I(sv+ + tv−) +(

12− 1

p

)‖u3‖2

≥ m +(

12− 1

p

)‖u3‖2.

which implies u3 = 0. Therefore, ũ has exactly two nodal domains.

4 Proof of Theorem 1.2

In view of Lemmas 2.8 and 2.9, there exist ū ∈ M such that

I(ū) = m, I′(ū) = 0. (4.1)

Furthermore, it follows from (1.2), (3.1), Corollary 2.3 and Lemma 2.7 that

m = I(ũ) = sups,t≥0

I(sũ+ + tũ−)

= sups,t≥0

[I(sũ+) + I(tũ−) +

bs2t2

2|∇ũ+|22|∇ũ|22

]≥ sup

s≥0I(sũ+) + sup

t≥0I(tũ−) ≥ 2c > 0.

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China(11571370, 11701487,11626202) and Hunan Provincial Natural Science Foundation of China(2016JJ6137). We would like to thank the anonymous referees for their valuable suggestionsand comments.


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IntroductionPreliminary lemmasProof of Theorem 1.1Proof of Theorem 1.2

ground state sign-changing solutions for kirchhoff equations with...

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