MULTIPLICITY OF SOLUTIONS TO KIRCHHOFF ...

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 17, Number 1, January 2018

doi:10.3934/cpaa.2018007 pp. 113–125

MULTIPLICITY OF SOLUTIONS TO KIRCHHOFF TYPE EQUATIONS WITH CRITICAL SOBOLEV EXPONENT

Peng Chen and Xiaochun Liu∗ School of mathematics and statistics, Wuhan University Wuhan, 430072, China

(Communicated by Yi Li) Abstract. This paper is concerned with the existence and multiplicity of solutions to the following Kirchhoff type elliptic equations with critical nonlinearity: ( R −(a + b Ω |∇u|2 dx)∆u = f (x, u) + µ|u|4 u in Ω, u=0

on ∂Ω,

R3

where Ω ⊂ is a bounded smooth domain, µ is a positive parameter and f : Ω × R → R is a Carath´ eodory function satisfying some further conditions. Our approach is based on concentration-compactness principle and symmetry mountain pass theorem.

1. Introduction and main results. In this paper, we study the existence and multiplicity of solutions to the following Kirchhoff type equation with critical nonlinearity: ( R −(a + b Ω |∇u|2 dx)∆u = f (x, u) + µ|u|4 u in Ω, (K) u=0 on ∂Ω, where Ω ⊂ R3 is a bounded smooth domain, a, b > 0, µ is a positive parameter and f : Ω × R → R is a Carath´eodory function satisfying sup {|f (x, s)| : x ∈ Ω, |s| ≤ T } < ∞ for every T > 0. Some further conditions will be stated R later on. Problem (K) is called nonlocal because of the presence of the term ( Ω |∇u|2 dx)∆u which implies that the equation in (K) is no longer a pointwise identity. This phenomenon provokes some mathematical difficulties, which makes the study particularly interesting. Problem (K) is a variant type of the following Dirichlet problem ( R −(a + b Ω |∇u|2 dx)∆u = g(x, u) in Ω, (1.1) u=0 on ∂Ω, which is related to the stationary analogue of the equation Z utt − (a + b |∇u|2 dx)∆u = g(x, u) Ω

2000 Mathematics Subject Classification. Primary: 35B33, 35J60; Secondary: 47J30. Key words and phrases. Kirchhoff type equation, multiplicity, critical nonlinearity. The authors are supported by NSFC grants 11371282 and 11571259. ∗ Corresponding author.

113

(1.2)

114

PENG CHEN AND XIAOCHUN LIU

proposed by Kirchhoff (see[9]) as an extension of the classical D’Alembert’s wave equation Z L 2 ! 2 ∂u ∂2u ρ0 E dx ∂ u = g(x, u) ρ 2 − (1.3) + ∂t h 2L 0 ∂x ∂x2 for free vibrations of elastic strings. Problem (1.2) began to attract the attention of several researchers mainly after the work of Lions [10], where a functional analysis approach was proposed to attack it. We have to point out that such nonlocal problems also appear in other fields as biological systems where u describes a process which depends on the average of itself (for example, population density). See, for example, [2] and references therein. Recently, Kirchhoff type problem (1.1) has been studied by many researchers. Zhang and Perera [19] studied sign changing solutions with subcritical growth. Mao and Zhang [13] considered sign changing and multiple solutions without PS condition later on. He and Zou [8] obtained infinitely many positive solutions with g(x, u) having an oscillating behavior which improves the main results in [19]. Existence and bifurcation of positive solution are considered by Liang et al. in [12]. Sun and Tang [17] also obtained existence and multiplicity of solutions with subcritical growth. Figueiredo et al. [5] got multiplicity of solutions with g(x, u) = λ|u|q−2 u + |u|p−2 u (1 < q < 2 < p ≤ 6) under a suitable range of the parameter λ. For other related results, we refer to [3, 6, 14, 18] and references therein. We point out that, except [5], the multiplicity results mentioned above are only considered in the subcritical case. Motivated by [16], we investigate the multiplicity of solutions to the problem (K) with critical term in this paper. A typical difficulty occurs in proving the existence of solutions. It is caused by the lack of compactness of the Sobolev embedding H01 (Ω) ,→ L6 (Ω). We successfully using the concentration compactness principle by Lions [11] to overcome this difficulty, though the nonlinear term f (x, s) satisfying some quite weak assumptions which has not been considered before. Before stating our main results, we Rgive the hypotheses on the function f : s Ω × R → R and its primitive F (x, s) := 0 f (x, t)dt as follows. (x,s) = 0 uniformly for x ∈ Ω. (f 1) There holds lim|s|→∞ f |s| 5 (f 2) There are θ ∈ (4, 6), σ ∈ (0, 2) and a1 , a2 > 0 such that 1 f (x, s)s − F (x, s) ≥ −a1 − a2 |s|σ for all s ∈ R, a.e. x ∈ Ω. θ (f 3) There are constants b1 , b2 > 0 and κ ∈ (2, 6) such that

F (x, s) ≤ b1 |s|κ + b2 for all s ∈ R, a.e. x ∈ Ω. (f 4) There exist c1 > 0, h1 ∈ L1 (Ω) and Ω0 ⊂ Ω with |Ω0 | > 0 such that lim inf |s|→∞

F (x, s) = ∞ uniformly for x ∈ Ω0 |s|4

and F (x, s) ≥ −h1 (x)|s|2 − c1 for all s ∈ R, a.e. x ∈ Ω. Observe that condition (f 2) is a weaker version of Ambrosetti-Rabinowitz condition [1]. Also, condition (f 4) is weak than the usual 4-superlinear at infinity. A typical example of a function satisfying the conditions (f 1)–(f 4) is given by f (x, u) = λ|u|p−2 u + |u|q−2 u with λ ∈ R, 1 < p < 2 and 4 < q < 6. Our first results is as follows.

KIRCHHOFF TYPE EQUATIONS WITH CRITICAL SOBOLEV EXPONENT

115

Theorem 1.1. Let a, b > 0. Suppose that f (x, s) is odd in s and satisfies (f 1)– (f 4). Then, for given k ∈ N, there exists µ∗ = µ∗ (k) > 0 such that problem (K) possesses at least k pairs of nontrivial solutions for all µ ∈ (0, µ∗ ). Remark 1. Under condition (f 1) and (f 2), the associated functional satisfies Palais-Smale condition below a fixed level when µ is sufficiently small (see Lemma 3.2). Instead of (f 3), if we assume (x,s) = a(x) λ1 uniformly for x ∈ Ω, where a(x) is a measu(f 5) lim sups→0 4Fb|s| 4 rable function, a(x) λ1 means that a(x) ≤ λ1 a.e. x ∈ Ω and a(x) < λ1 on a set of positive measure, λ1 is the principle eigenvalue of the nonlocal eigenvalue problem (EP) (see definition in section 2), then we have:

Theorem 1.2. Let a ≥ 0 and b > 0. Suppose that f (x, s) is odd in s and satisfies (f 1), (f 2), (f 4), (f 5). Then, given k ∈ N, there exists µ∗ = µ∗ (k) > 0 such that problem (K) possesses at least k pairs of nontrivial solutions for all µ ∈ (0, µ∗ ). Remark 2. Condition (f 5) is stronger than (f 3) in the sense that it provides a better symmetric mountain pass geometry (see Lemma 5.2 and Lemma 4.2). Instead of (f 4), if we assume (f 6) there exist constants B > b and C > 0 such that B λ1 |s|4 − C for all s ∈ R, a.e. x ∈ Ω, 4 where λ1 is the same as in (f 5), then we have: F (x, s) ≥

Theorem 1.3. Let a ≥ 0 and b > 0. Suppose that f (x, s) satisfies f (x, 0) = 0, (f 1), (f 2), (f 5), (f 6). Then, there exists µ∗ ∈ (0, ∞) such that (K) possesses a positive and a negative solution for all µ ∈ (0, µ∗ ). This paper is organized as follows. In Section 2, we provide some preliminaries and notations. Section 3 is devoted to verify the Palais-Smale condition. In section 4, 5 and 6, we prove Theorem 1.1, 1.2, 1.3, respectively. 2. Preliminaries and notations. Recall that u ∈ H01 (Ω) is a weak solution of the problem (K) if it satisfies Z Z Z Z (a + b |∇u|2 dx) ∇u∇ϕdx = f (x, u)ϕdx + µ |u|4 uϕdx Ω

Ω

Ω

Ω

H01 (Ω).

for all ϕ ∈ We look for solutions of (K) by searching critical points of the C 1 -functional Iµ : H01 (Ω) → R given by Z a b µ 2 4 Iµ (u) = kuk + kuk − F (x, u)dx − |u|66 2 4 6 Ω R R 1 1 where kuk = ( Ω |∇u|2 dx) 2 and |u|p = ( Ω |u|p dx) p are the standard norms in H01 (Ω) and Lp (Ω), respectively. By (f 1) and sup{|f (x, s)| : x ∈ Ω, |s| ≤ T } < ∞ for every T > 0, we have that for any > 0 there exists a constant C > 0 such that |f (x, s)s| ≤ C + |s|6

(2.1)

116


and

|F (x, s)| ≤ C + |s|6 (2.2) 6 for every s ∈ R, a.e. x ∈ Ω. The estimate (2.1) and (2.2) will be used frequently in the context. Next we introduce a eigenvalue problem. Let λ1 be the principle eigenvalue of the following eigenvalue problem ( −kuk2 ∆u = λu3 in Ω, (EP) u=0 on ∂Ω. In [19], Zhang and Perera proved that λ1 > 0 and the corresponding eigenfunction φ1 > 0 in Ω. Moreover we have the following characterization: Z 4 1 4 λ1 = inf kuk : u ∈ H0 (Ω) and |u| dx = 1 . Ω

Since H01 (Ω) is a separable Hilbert {ei } ⊂ H01 (Ω)(see [7]). Then, for any u u=

space, we can choose a orthonormal basis ∈ H01 (Ω), we have

∞ X

hu, ei iei .

(2.3)

i=1

We set, for j ∈ N, Vj = {u ∈ H01 (Ω) : hu, ei i = 0, i > j},

(2.4)

Xj = {u ∈ H01 (Ω) : hu, ei i = 0, i ≤ j}.

It follows by (2.3) that H01 (Ω) = Vj ⊕ Xj and dim Vj = j. To prove our main results, we will use the following versions of mountain pass theorem. Theorem 2.1 ([16]). Let E = V ⊕ X, where E is a real Banach space and V is finite dimensional. Suppose I ∈ C 1 (E, R) is an even functional satisfying I(0) = 0 and (i) there are constants ρ, α > 0 such that I |∂Bρ ∩X ≥ α; (ii) there exists a subspace W of E with dim V < dim W < ∞ such that max I(u) < u∈W

M for some constant M > 0; (iii) I satisfies (P S)c for 0 < c < M , where M is given by (ii). Then I possesses at least dim W − dim V pairs of nontrivial critical points. Theorem 2.2 ([16]). Let E be a real Banach space. Suppose I ∈ C 1 (E, R) satisfies I(0) = 0 and (i) there are constants ρ, α > 0 such that I |∂Bρ ≥ α; (ii) there exists v1 ∈ ∂B1 (0) such that sup I(tv1 ) ≤ M for some constant M > 0; t≥0

(iii) I satisfies (P S)c for 0 < c < M , where M is given by (ii). Then I possesses a nontrivial critical point. Throughout this paper, we denote B(x, r) ⊂ R3 , Bρ = Bρ (0) ⊂ H01 (Ω) as an open ball, respectively, |Ω| as the Lebesgue measure for an arbitrary subset Ω ⊂ R3 and C, C 0 , C > 0 as any positive constants.


117

3. Palais-Smale condition. In this section, we verify that the functional Iµ satisfies the (P S)c condition below a given level when µ > 0 is sufficiently small. To do this, we begin with some convergence result. Lemma 3.1. Suppose that f satisfies (f 1). Let {un } ⊂ H01 (Ω) be a bounded sequence. Then, there exists a function u ∈ H01 (Ω) such that, up to a subsequence, Z lim |f (x, un )un − f (x, u)u|dx = 0, (3.1) n→∞ Ω Z Z lim f (x, un )vdx = f (x, u)vdx, (3.2) n→∞

Ω

Ω

and Z lim

n→∞

|un |4 un vdx =

Ω

Z

|u|4 uvdx,

(3.3)

Ω

for every v ∈ H01 (Ω). Proof. It suffices to prove (3.1), since (3.2) and (3.3) can be obtained in a similar way. By the boundedness of {un }, there exist a function u ∈ H01 (Ω) and a constant C > 0 such that |un |66 ≤ C, un * u weakly in H01 (Ω), un → u a.e. in Ω. By (2.1), for any > 0 there exists a constant C > 0 such that |f (x, s)s| ≤ C + Set δ =

2C .

|s|6 , for all s ∈ R, a.e. x ∈ Ω. 2C

Then for any Ω0 ⊂ Ω with |Ω0 | < δ, we have Z

Ω0

Z f (x, un )un dx ≤

|f (x, un )un | dx Z Z ≤ C dx + |un |6 dx 2C Ω0 Ω0 < + = . 2 2 Ω0

Hence {f (x, un )un }n∈N is uniformly integrable. By Vitali’s convergence theorem, we immediately obtain our conclusion. Lemma 3.2. Suppose that f satisfies (f 1) and (f 2). Then, for given M > 0, there exists µ∗ > 0 such that Iµ satisfies (P S)c condition for all c < M , provided 0 < µ < µ∗ . Proof. Given M > 0, let {un } ⊂ H01 (Ω) be a (P S)c sequence for Iµ with c < M . We divide our proof into three steps. Step 1. We claim that {un } is bounded in H01 (Ω). Since Iµ (un ) → c and Iµ0 (un ) → 0 in H −1 (Ω), by (f 2), for n sufficiently large,

118


we have 1 1 c + 1 ≥ Iµ (un ) − hIµ0 (un ), un i + hIµ0 (un ), un i θ θ Z 1 1 1 1 1 1 2 4 ≥ a( − )kun k + b( − )kun k + µ( − ) |un |6 dx 2 θ 4 θ θ 6 Ω Z 1 f (x, un )un − F (x, un ) dx − kun k + θ Ω 1 1 ≥ µ( − )|un |66 − a1 |Ω| − a2 |un |σσ − kun k θ 6 1 1 1 ≥ µ( − )|un |66 − kun k − a1 |Ω| − C, 2 θ 6

(3.4)

where C is some positive constant depending only on µ, θ, a2 and |Ω|. In the last inequality of (3.4), we have used the H¨older and Young inequality as follows 6(1−α)

|un |σσ ≤ |Ω|α |un |6 6(1−α)

|un |6 with α =

6−σ 6 ,

δ=

µ( θ1 − 16 ) 2a2 |Ω|α

,

≤ δ|un |66 + Cδ ,

(1−α)/α and Cδ = α( 1−α . Thus from (3.4) we get δ )

|un |66 ≤ C + Ckun k.

(3.5)

By (2.2) and (3.5), there exists a constant C > 0 such that Iµ (un ) ≥

a b kun k2 + kun k4 − Ckun k − C. 2 4

(3.6)

Since Iµ (un ) is bounded, we conclude that {un } is bounded in H01 (Ω). Hence by the weak compactness of H01 (Ω) and the compactness of the Sobolev embedding, there exists a function u ∈ H01 (Ω) such that un * u weakly in H01 (Ω), un → u in Lp (Ω) for all 1 ≤ p < 6, un → u a.e. on Ω, up to subsequences but still denoted by {un }. Moreover from the second concentration compactness lemma by Lions [11], there exist an at most countable set J , points {xk }k∈J ⊂ Ω and numbers {ηk }k∈J , {νk }k∈J ⊂ R+ such that, up to subsequences, X |∇un |2 * dη ≥ |∇u|2 + η k δ xk , k∈J 6

6

|un | * dν = |u| +

X

νk δxk ,

k∈J

in the sense of measures. Here δx is the Dirac measure concentrated at x ∈ R3 with mass 1. In addition, we also have the inequality 1

ηk ≥ Sνk3 . Step 2. We claim νk ≥

bS µ

3 2

(3.7)

if J 6= ∅.

Suppose J = 6 ∅. Fix k ∈ J . Define a smooth function φ such that φ = 1 on


119

B(xk , ), φ = 0 on B(xk , 2)c and 0 ≤ φ ≤ 1 otherwise. Moreover we can assume |∇φ| ≤ 2/. As I 0 (un ) → 0 in H −1 (Ω), we have 0 = lim hIµ0 (un ), un φi n→∞ Z Z 2 = lim (a + bkun k ) ∇un · ∇(un φ)dx − f (x, un )un φdx n→∞ Ω Ω Z − lim µ |un |6 φdx n→∞ Ω Z Z = lim (a + bkun k2 ) |∇un |2 φdx − µ |un |6 φdx + o(1), n→∞

Ω

Ω

where o(1) → 0 as → 0. The last equality comes from the facts that Z lim (a + bkun k2 ) (∇un · ∇φ)un dx = o(1) as → 0, n→∞

(3.8)

(3.9)

Ω

and

Z f (x, un )un φdx = o(1) as → 0.

lim

n→∞

(3.10)

Ω

We first verify (3.9). By the boundedness and L2 (Ω)-convergence of {un } and the H¨ older inequality, we have Z lim (a + bkun k2 ) (∇un · ∇φ)un dx n→∞

Ω

Z ≤C lim

n→∞

12 12 Z 2 2 |un | |∇φ| dx |∇un | dx 2

Ω

Ω

! 16

Z

6

≤C

! 13

Z

3

|u| dx

|∇φ| dx

Ω∩B(xk ,2)

Ω∩B(xk ,2)

! 16

Z

|u|6 dx

≤C

→ 0 as → 0,

Ω∩B(xk ,2)

where we use our assumption |∇φ| ≤ 2/. For (3.10), by (3.1) and 0 ≤ φ ≤ 1, we obatin Z Z lim f (x, un )un φdx = f (x, u)uφdx = o(1) as → 0. n→∞

Ω

Ω∩B(xk ,2)

From (3.8), we have Z Z Z 0≥ a+b φdη φdη − µ φdν + o(1) Ω

Ω

as → 0.

Ω

Taking → 0 above, we get 0 ≥ (a + bηk )ηk − µνk . Combining (3.7) with (3.11), we estimate !3 p 3 bS 2 + (bS 2 )2 + 4aµS bS 2 ≥ νk ≥ 2µ µ

(3.11)

(3.12)

as required. Step 3. There exists µ∗ > 0 such that Iµ satisfies (P S)c condition for all c < M provided 0 < µ < µ∗ .

120


Given M > 0, set

µ∗ = min

  

 bS 2 ,  bS 2

3

  where A = a1 |Ω| + a2 |Ω|α , α = 0 < µ < µ∗ , we have

6−σ 6

1 θ

1 6

− M +A

α1

 1  3−1/α     

(3.13)

and a1 , a2 , σ are the constants in (f 2). For

1
0, j ∈ N and ρ, α > 0 such that Iµ |∂Bρ ∩Xj ≥ α for all 0 < µ < µ e. Proof. By (f 3), Sobolev embedding theorem and Lemma 4.1, there exist b3 , δ > 0 such that Z µ a 2 F (x, u)dx − |u|66 Iµ (u) ≥ kuk − 2 6 Ω a ≥ kuk2 − b1 |u|κκ − b2 |Ω| − b3 µkuk6 2 a ≥kuk2 − b1 δkukκ−2 − b2 |Ω| − b3 µkuk6 2 for all u ∈ Xj and j sufficiently large. Take ρ = ρ(δ) with b1 δρκ−2 = a/4. Noting that ρ(δ) → ∞ as δ → 0, we choose δ > 0 sufficiently small such that a4 ρ2 − b2 |Ω| > a 2 e > 0 such that α := a8 ρ2 − b3 µ eρ6 > 0. Then, for any u ∈ Xj 8 ρ . Next we take µ with ||u|| = ρ and 0 < µ < µ e, we have a a eρ6 = α. Iµ (u) ≥ ρ2 − b2 |Ω| − b3 µρ6 ≥ ρ2 − b3 µ 4 8 The proof is completed. Lemma 4.3. Suppose f satisfies (f 4). Then, given m ∈ N, there exists a subspace W of H01 (Ω) with dim W = m such that maxu∈W I0 (u) < M for some M > 0. Proof. We follow the argument of Lemma 4.3 in [16]. Take x0 ∈ Ω0 and r0 > 0 such that B(x0 , r0 ) ⊂ Ω and 0 < |B(x0 , r0 ) ∩ Ω0 | < |Ω0 |/2. First, we choose c0 = v1 ∈ C0∞ (Ω) with suppv1 = B(x0 , r0 ). Let Ω1 = Ω0 \ (B(x0 , r0 ) ∩ Ω0 ) ⊂ Ω Ω \ B(x0 , r0 ), we have |Ω1 | > |Ω0 |/2 > 0. Take x1 ∈ Ω1 and r1 > 0 such that c0 and 0 < |B(x1 , r1 ) ∩ Ω1 | < |Ω1 |/2. Next, we choose v2 ∈ C ∞ (Ω) B(x1 , r1 ) ⊂ Ω 0 ∞ with suppv2 = B(x1 , r1 ). After a finite number of steps, we get {vi }m i=1 ⊂ C0 (Ω) with suppvi ∩ suppvj = B(xi , ri ) ∩ B(xj , rj ) = ∅, i 6= j and |suppvj ∩ Ω0 | > 0 for all i, j ∈ {1, . . . , m}. Let W = span{v1 , . . . , vm }. Since {vi }m i=1 are mutually orthogonal in H01 (Ω), we have dim W = m. Moreover we have Z |v|4 dx > 0 for every v ∈ W \ {0}. (4.1) Ω0

Set Z R1 = min

|v|4 dx : v ∈ ∂B1 (0) ∩ W

Ω0

and R2 = max |v|2∞ : v ∈ ∂B1 (0) ∩ W .

122


Since W is finite dimensional, we have R1 > 0, R2 < ∞ and max I0 (u) =

u∈W

max

t>0 v∈∂B1 (0)∩W

I0 (tv).

By (f 4), given L > 0, there exists C > 0 such that F (x, s) ≥ L|s|4 − C for all s ∈ R, a.e. x ∈ Ω0 .

(4.2)

b 4R1 .

By (4.2) and (f 4), for any v ∈ ∂B1 (0) ∩ W and t > 0, Z a 2 b 4 I0 (tv) = t + t − F (x, tv)dx 2 4 Ω Z Z b a |v|4 dx + C|Ω0 | + t2 |h1 ||v|2 dx + c1 |Ω \ Ω0 | ≤ t2 + t4 − Lt4 2 4 Ω0 Ω\Ω0 a 2 b 4 4 2 ≤ t + t − LR1 t + |h1 |1 R2 t + C → −∞, as t → ∞. 2 4 Thus there exists M > 0 such that maxu∈W I0 (u) < M . The proof is completed. Take L >

Now we can prove Theorem 1.1. Proof of Theorem 1.1. Recall that H01 (Ω) = Vj ⊕ Xj with dim Vj = j where Vj and Xj are defined in (2.4). Note that Iµ (0) = 0 and Iµ is even. Given k ∈ N, we set m = j + k and µ∗ (k) = min{µ∗ , µ e} where j, µ e and µ∗ are provided by Lemma 4.2 and Lemma 3.2, respectively. By Lemma 4.2, Lemma 4.3 and Lemma 3.2, Iµ satisfies (i), (ii) and (iii) of Theorem 2.1 for all 0 < µ < µ∗ (k). Thus, Iµ possesses at least dim W −dim Vj = k pairs of nontrivial critical points for all µ ∈ (0, µ∗ (k)). 5. Proof of Theorem 1.2. In this section, we prove Theorems 1.2 by verifying that the functional Iµ satisfies the hypotheses of Theorem 2.1. Lemma 5.1. Let a(x) : Ω → R be a measurable function such that a(x) λ1 . Then there exists β > 0 such that Z Z kuk4 − a+ (x)|u|4 dx ≥ β |u|4 dx for every u ∈ H01 (Ω), Ω

Ω

+

where a (x) = max{a(x), 0}. Proof. Arguing by contradiction, we suppose that for every n ∈ N there exists a un ∈ H01 (Ω), such that Z Z 1 kun k4 − a+ (x)|un |4 dx < |un |4 dx. n Ω Ω Let vn =

un |un |4 .

Since a(x) ≤ λ1 , we have Z 1 1 λ1 ≤ kvn k4 < a+ (x)|vn |4 dx + ≤ λ1 + . n n Ω

In particular, {vn } is bounded in H01 (Ω). Therefore, up to subsequence, vn * v weakly in H01 (Ω), vn → v in L4 (Ω), vn → v a.e. on Ω, |vn (x)| ≤ h(x) ∈ L4 (Ω) a.e. x ∈ Ω.

(5.1)


123

Thus |v|4 = 1. Taking n → ∞ in (5.1) and applying the Lebesgue’s dominated convergence theorem, we get Z (λ1 − a+ (x))|v|4 dx = 0. (5.2) Ω

Furthermore, by (5.1) and the characterization of λ1 , we have kvk4 ≤ lim inf kvn k4 = λ1 ≤ kvk4 . n→∞

Hence, v is an eigenfunction associated with λ1 and v > 0 in Ω. This contradicts to (5.2) and a(x) λ1 . The proof is completed. Lemma 5.2. Suppose f satisfies (f 1) and (f 5). Given µ > 0, there are constants ρ, α > 0 such that Iµ |∂Bρ ≥ α. Proof. By (f 5) and sup{|f (x, s)| : x ∈ Ω, |s| ≤ T } < ∞ for every T > 0, for any > 0 there exists a constant C > 0 such that b C 6 (a(x) + )|s|4 + |s| for every s ∈ R, a.e. x ∈ Ω. (5.3) 4 6 Take 0 > 0 such that β − 0 λ1 > 0 where β is provided by Lemma 5.1. Since a+ (x) ≤ λ1 a.e. x ∈ Ω, we have Z Z 1 4 4 0 4 + 4 kuk − a(x)|u| dx ≥ (1 + ) kuk − a (x)|u| dx 1 + 0 Ω Ω Z Z 1 4 0 4 4 ≥ β |u| dx + kuk − λ1 |u| dx 1 + 0 (5.4) Z Ω Ω 1 0 4 0 4 = (β − λ1 )|u| dx + kuk 1 + 0 Ω 0 kuk4 . ≥ 1 + 0 F (x, s) ≤

Combining (5.3), (5.4) and the Sobolev embedding theorem, we obatin Z Z a b µ Iµ (u) = kuk2 + kuk4 − F (x, u)dx − |u|6 dx 2 4 6 Ω Ω Z Z Z b b 4 4 |u|4 dx − C |u|6 dx ≥ kuk − a(x)|u| dx − 4 4 Ω Ω Ω Z b0 b 4 ≥ kuk − |u|4 dx − Ckuk6 4(1 + 0 ) 4 Ω ≥ C 0 kuk4 − Ckuk6 for > 0 sufficiently small. Taking ρ > 0 small enough, we conclude that there exists a constant α > 0 such that Iµ (u) ≥ α with kuk = ρ. Now we can prove Theorem 1.2. Proof of Theorem 1.2. Note that H01 (Ω) = {0} ⊕ H01 (Ω), Iµ (0) = 0 and Iµ is even. Given k ∈ N, denote µ∗ (k) = µ∗ where µ∗ is provided by Lemma 3.2. By Lemma 5.2, Lemma 4.3 and Lemma 3.2, Iµ satisfies (i), (ii) and (iii) of Theorem 2.1 for all 0 < µ < µ∗ (k). Thus, Iµ possesses at least dim W − dim{0} = k pairs of nontrivial critical points for all µ ∈ (0, µ∗ (k)).

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6. Proof of Theorem 1.3. In this section, we prove Theorems 1.3. Proof of Theorem 1.3. It suffices to prove the existence (K). We consider the following equivalent problem  R 2 5 e  −(a + b Ω |∇u| dx)∆u = f (x, u) + µu u>0   u=0 where

( fe(x, s) =

f (x, s) 0

of a positive solution of in Ω, in Ω, on ∂Ω,

(6.1)

if s > 0, if s ≤ 0.

The energy functional Ieµ ∈ C 1 (H01 (Ω), R) associated with (6.1) is given by Z Z µ a b 2 4 e e F (x, u)dx − Iµ (u) = kuk + kuk − (u+ )6 dx, 2 4 6 Ω Ω Rs where Fe(x, s) = 0 fe(x, t)dt. We will show that Ieµ possesses a nontrivial critical point. Note that by Lemma 5.2, Ieµ satisfies (i) of Theorem 2.2. Now take φ1 > 0 be the eigenfunction associated with λ1 . By (f 6), for every t > 0, we have Z a 2 b 4 B 4 2 4 e Iµ (tφ1 ) ≤ t kφ1 k + t kφ1 k − t λ1 |φ1 |4 dx + C|Ω| → −∞ 2 4 4 Ω as t → ∞. Hence supt>0 Ieµ (tφ1 ) < M for some M > 0 and Ieµ satisfies (ii) of Theorem 2.2. Taking {un } ⊂ H 1 (Ω) with Ieµ (un ) → c and Ieµ0 (un ) → 0 in H −1 (Ω), 0

1 it is easy to see that u− n → 0 in H0 (Ω) as n → ∞. Indeed, by analogous argument as in the proof of Lemma 3.2, we have {un } is bounded. Note that (a + bkun k2 )ku− k2 = hIe0 (un ), u− i ≤ kIe0 (un )kku− k. n

Since

Ieµ0 (un )

→ 0 in H

−1

µ

n

µ

n

(Ω), this gives us the assertion. Hence, we have

e + e Iµ (u+ n ) = Iµ (un ) = Iµ (un ) − o(1), e0 + e0 Iµ0 (u+ n ) = Iµ (un ) = Iµ (un ) − o(1). Therefore, {u+ n } is a (P S)c sequence for Iµ . By Lemma 3.2, there exists µ∗ > 0 − + such that {u+ n } has a convergent subsequence with µ ∈ (0, µ∗ ). Since un = un − un , 1 e u− n → 0 in H0 (Ω), we conclude {un } has a convergent subsequence. Thus Iµ satisfies e (iii) of Theorem 2.2 with µ ∈ (0, µ∗ ). By Theorem 2.2, Iµ possesses a nontrivial critical point u ∈ H01 (Ω). Obviously, u ≥ 0. By strong maximum principle, u > 0 in Ω. This completes the proof. Acknowledgments. The authors wish to thank the anonymous referees very much for carefully reading this paper and suggesting many valuable comments. REFERENCES [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349–381. [2] C. O. Alves, F. J. S. A. Corra and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85–93. [3] C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 23 (2010), 409–417. [4] H. Br´ ezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490.


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Received January 2017; revised July 2017. E-mail address: [email protected] E-mail address: [email protected]