Multiple solutions for a class of fractional equations

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Multiple solutions for a class of fractional equations

Ruichang Pei

^B¹

, Jihui Zhang

²

and Caochuan Ma

¹

1School of Mathematics and Statistics Tianshui Normal University, Tianshui, 741001, P. R. China

2Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing, 210097, P. R. China

Received 8 October 2015, appeared 28 December 2015 Communicated by Dimitri Mugnai

Abstract. In this paper we study a class of fractional Laplace equations with asymptotically linear right-hand side. The existence results of three nontrivial solutions under the resonance and non-resonance conditions are established by using the minimax method and Morse theory.

Keywords: fractional Laplacian, multiple solutions, asymptotically linear, mountain pass theorem, Morse theory.

2010 Mathematics Subject Classification: 35J60, 49J35.

1 Introduction

In this article, we are interested in the following non-local fractional equations:

((−_∆)^su= f(x,u), inΩ,

u=0, inR^N\_Ω, ^(1.1)

where s ∈ (0, 1) is a fixed parameter,Ω is a bounded domain inR^N with smooth boundary

∂Ω,N >2sand(−_∆)^sis the fractional Laplace operator.

In recent years, many papers are devoted to the study of non-local fractional Laplacian with superlinear and subcritical or critical growth (see [2,4,13,14,16,17] and references therein).

Particularly, in [8], Fiscella et al. studied equations (1.1) with asymptotically linear right-hand side and obtained some existence results by using saddle point theorem; in [9], Iannizzotto et al. also studied fractional p-Laplacian equations with asymptotically p-linear and obtained two nontrivial solutions by the use of the mountain pass theorem.

There are many interesting problems in the standard framework of the Laplacian (or higher order Laplacian), widely studied in the literature. A natural question is whether or not the existence results of multiple solutions obtained in the classical context can be extended to the non-local framework of the fractional Laplacian operator. Chang et al. [7] showed the existence of three nontrivial solutions for asymptotically linear Dirichlet problem via the mountain pass

BCorresponding author. Email: prc211@163.com

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theorem and Morse theory. In [11] Qian et al. did similar work for fourth-order asymptotically linear elliptic problem.

Motivated by their work, we study the following non-local problem with homogeneous Dirichlet boundary conditions investigated by Servadei et al. [15] and the related works [12, 14]:

(−L_ku= _f(_x,_u)_, _in_Ω,

u=0, inR^N\_Ω, ^(1.2)

whereL_k is the integro-differential operator defined as follows:

L_ku(x) =

Z

R^N(u(x+y) +u(x−y)−2u(x))K(y)dy, x∈_R^N, (1.3) with the kernelK:R^N\0→(0,+_∞)such that

(B1) mK∈ L¹(_R^N)_{, where}m(x) =_min{|x|²_{, 1}}_,

(B2) there existsθ>0 such thatK(x)≥ θ|x|⁻⁽^N⁺^2s⁾ for any x∈_R^N\{0}, (B3) K(x) =K(−x)for any x∈_R^N\{0}.

For narrative convenience, in this paper, we only consider the particular case of problem (1.2), i.e., we let K be given by the singular kernel K(x) = |x|⁻⁽^N⁺^2s⁾ which leads to the fractional Laplace operator−(−_∆)^s, which, up to normalization factors, may be defined as

−(−_∆)^su(x) =

Z

R^N

u(x+y) +u(x−y)−2u(x)

|y|^N⁺^2s ^dy, ^x∈_R^N. (1.4) Obviously, the corresponding fractional equation in the above model (1.2) changes problem (1.1). In fact, our methods and results in this paper also adapt for the general problem (1.2).

Let f(x, 0) =0 andF(x,t) =Rt

0 f(x,s)ds. Moreover, suppose that the non-linearity f satisfy the following conditions:

(f₁) f ∈C¹(_Ω^¯ ×_R,_R), f(x, 0) =0, f(x,t)t≥0 for all x∈_Ω,t ∈_R,

(f₂) f⁰ is subcritical int, i.e. there is a constantp∈ (2, 2^∗), 2^∗ = _N^2N₋_2s such that

tlim→_∞

f_t(x,t)

|t|^p⁻¹ =₀ uniformly forx ∈_Ω,^¯ (f₃) lim

|t|→0 f(x,t)

t = f₀, lim

|t|→_∞ f(x,t)

t =luniformly for x∈_Ω, where f₀ andlare constants;

(f₄) lim

|t|→_∞[f(x,t)t−2F(x,t)] =−_∞. Now, we give our main results.

Theorem 1.1. Assume conditions (f₁)–(f3)hold, f0 < λ₁ and l ∈ (λ_k,λ_k+1)for some k ≥ 2, then problem(1.1)has at least three nontrivial solutions.

Theorem 1.2. Assume conditions (f₁)–(f₄)hold, f₀ < λ₁ and l = λ_k for some k ≥ 2, then problem (1.1)has at least three nontrivial solutions.

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Here, 0 < λ₁ < λ₂ < · · · < λ_k < · · · are the eigenvalues of (−_∆)^s with homogeneous Dirichlet boundary data and φ₁(x)>0 be the eigenfunction corresponding toλ₁.

In view of the condition(f₃), problem (1.1) is called asymptotically linear at both zero and infinity, which means that usual Ambrosetti–Rabinowitz condition (see [1]) is not satisfied.

This will bring some difficulty if the mountain pass theorem is used to seek nontrivial solutions of problem (1.1). For the standard Laplacian Dirichlet problem, Zhou [18] have overcome it by using some monotonicity condition. Novelties of our this paper are as following.

We consider multiple solutions of problem (1.1) in the cases of resonance and non- resonance by using the mountain pass theorem and Morse theory. First, we use the trun- cated technique and the mountain pass theorem to obtain a positive solution and a negative solution of problem (1.1) under our more general conditions (f₁), (f₂) and(f₃)with respect to the conditions (H₁)and (H₃) in [18]. In the course of proving the existence of a positive solution and a negative solution, the monotonicity condition (H2) of [18] on the nonlinear term f is not necessary, this point is very important because we can directly prove existence of positive solution and negative solution by using Rabinowitz’s mountain pass theorem. That is, the proof of our compact condition is more simple than that in [18]. Furthermore, we can obtain a nontrivial solution when the nonlinear term f is resonant or non-resonant at the infinity by using Morse theory.

The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge about the working space. In Section 3, we prove some lemmas in order to prove our main results. In Section 4, we give the proofs for our main results.

2 Preliminaries

In this section, we give some preliminary results which will be used in the sequel. We briefly recall the related definitions and notes for functional spaceX0introduced in [15].

The functional space X denotes the linear space of Lebesgue measurable functions from R^N to Rsuch that the restriction toΩof any function gin X belongs to L²(_Ω)and the map (x,y)7−→(g(x)−g(y))^pK(x−y)is inL²(_R^N×_R^N)\(C_Ω× C_Ω),dxdy)(hereC_Ω=_R^N\_Ω). Also, we denote byX₀the following linear subspace of X

X0:={g ∈X: g=0 a.e. inR^N\_Ω}.

Note that X and X₀ are non-empty, since C₀²(_Ω) ⊆ X₀ by [15]. Moreover, the space X is endowed with the norm defined as

kgk_X= |g|_L2(_Ω)+ _Z

Q

|g(x)−g(y)|²K(x−y)dxdy ¹₂

, (2.1)

whereQ= (_R^N×_R^N)\O_andO= (C_Ω)×(C_Ω)⊂_R^N×_R^N_{. We equip}X₀with the following norm

kgk_X₀ = _Z

Q

|g(x)−g(y)|²K(x−y)dxdy ¹₂

, (2.2)

which is equivalent to the usual one defined in (2.1) (see [14]). It is easy to see that(X0,k · k_X₀) is a Hilbert space with scalar product

hu,vi_X₀ =

Z

Q

(u(x)−u(y))(v(x)−v(y))K(x−y)dxdy. (2.3)

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Denote byH^s(_Ω)the usual fractional Sobolev space with respect to the Gagliardo norm kgk_Hs(_Ω)=|g|_L2(_Ω)+

_Z

Ω×_Ω

|g(x)−g(y)|²

|x−y|^N⁺^2s ^dxdy ¹₂

. (2.4)

Now, we give a basic fact which will be used later.

Lemma 2.1 ([14]). The embedding j : X₀ ,→ L^v(_Ω) is continuous for any v ∈ [1, 2^∗], while it is compact whenever v∈[_{1, 2}^∗)_.

Next, we state some propositions for the operator (−_∆)^s. Let λ₁ < λ₂ ≤ λ₃ ≤ · · · ≤ λ_k ≤ · · · be the sequence of the eigenvalues of(−_∆)^s(see [8]) andφ_kbe thek-th eigenfunction corresponding to the eigenvaluesλ_k. Moreover, we will set

Pk+1 ={u∈X₀ :hu,φ_ji_X₀ =0, ∀j=1, 2, . . . ,k} and

H_k =_span{φ₁, . . . ,φ_k}_. Proposition 2.2([8]). The following inequality holds true

kuk²_X

0 ≤λ_k|u|²_L2(_Ω)

for all u∈H_k and k∈N.

Proposition 2.3([8]). The following inequality holds true kuk²_X

0 ≥λ_k+1|u|²_L2(_Ω)

for all u∈_P_k₊₁ and any k∈ N.

Next, we recall some definitions for compactness condition and a version of the mountain pass theorem.

Definition 2.4. Let (X₀,k · k_X₀) be a real Banach space with its dual space (X₀^∗,k · k_X^∗

0) and

J ∈ C¹(X0,R). For c ∈ R, we say that J satisfies the (PS)_c condition if for any sequence {x_n} ⊂X₀ with

J(x_n)→c, DJ(x_n)→0 inX₀^∗,

there is a subsequence {x_n_k} such that{x_n_k}converges strongly in X₀. Also, we say that J satisfy the(C)_c condition stated in [5] if for any sequence{x_n} ⊂X₀ with

J(xn)→c, kDJ(xn)k_X^∗

0(1+kxnk_X₀)→0, there is subsequence{x_n_k}such that{x_n_k}converges strongly inX₀.

3 Some lemmas

First, we observe that problem (1.1) has a variational structure, indeed it is the Euler–Lagrange equation of the functionalJ :X₀→_Rdefined as follows:

J(u) = ¹ 2 Z

R^N×_R^N|u(x)−u(y)|²K(x−y)dxdy−

Z

ΩF(x,u(x))dx.

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It is well know that the functionalJ is Fréchet differentiable in X₀and for any ϕ∈ X₀ hJ⁰(u),ϕi=

Z

R^N×_R^N

(u(x)−u(y))(ϕ(x)−ϕ(y))K(x−y)dxdy−

Z

Ω f(x,u(x))ϕ(x)dx.

Thus, critical points of J are solutions of problem (1.1).

Consider the following problem

((−_∆)^su= f+(x,u), inΩ;

u=0, inR^N\_Ω,

where

f+(x,t) =

(f(x,t), t >0,

0, t ≤_0.

Define a functionalJ+:X₀ →_Rby J₊(u) = ¹

2 Z

R^N×_R^N|u(x)−u(y)|²K(x−y)dxdy−

Z

ΩF(x,u(x))dx, where F+(x,t) =Rt

0 f+(x,s)ds, thenJ+∈C²⁻⁰(X0,R). Lemma 3.1. J+satisfies the (PS) condition.

Proof. Let{u_n} ⊂ X₀ be a sequence such that|J₊⁰(u_n)| ≤c,hJ₊⁰(u_n),ϕi →0 asn →_{∞. Note} that

hJ₊⁰(u_n),ϕi=

Z

R^N×_R^N(u_n(x)−u_n(y))(ϕ(x)−ϕ(y))K(x−y)dxdy−

Z

Ω f+(x,u_n)ϕdx

=o(kϕk_X₀)

(3.1)

for all ϕ ∈ X₀. Assume that |u_n|_L2(_Ω) is bounded, taking ϕ = u_n in (3.1). By (f₃), there exists c > 0 such that |f+(x,u_n(x))| ≤ c|u_n(x)|, a.e. x ∈ _Ω. So u_n is bounded in X₀. If

|un|_L2(_Ω) → +_{∞, as} n → _{∞, set} vn = _|_u ^uⁿ

n|_L₂₍_Ω₎, then |vn|_L2(_Ω) = 1. Taking ϕ = vn in (3.1), it follows thatkv_nk_X₀ is bounded. Without loss of generality, we assume thatv_n*vinX₀, then vn→vin L²(_Ω). Hence,vn→va.e. in Ω. Dividing both sides of (3.1) by|un|_L2(_Ω), we get

Z

R^N×_R^N

(_v_n(_x)−vn(_y))(ϕ(_x)−ϕ(_y))_K(_x−y)_dxdy−

Z

Ω

f+(x,u_n)

|un|_L2(_Ω)

ϕdx

=o kϕk_X₀

|un|_L2(_Ω)

!

, ∀ϕ∈X₀.

(3.2)

Then for a.e.x ∈ Ω, we deduce that _|^f_u⁺_n⁽_|^x,uⁿ⁾

L2(_Ω)

→lv+ asn→_{∞, where}v+=max{v, 0}. In fact, whenv(x)>0, by(f₃)we have

u_n(x) =v_n(x)|u_n|_L2(_Ω) →+_∞

and f+(x,u_n)

|un|_L2(_Ω)

= ^f⁺(x,u_n) un

v_n→lv.

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Whenv(x) =0, we have

f+(x,un)

|u_n|_L2(_Ω)

≤c|vn| −→0.

Whenv(_x)<_{0, we have}

un(x) =vn(x)|un|_L2(_Ω) −→ −_∞

and f+(x,un)

|u_n|_L2(_Ω)

=0.

Since _|^f_u⁺⁽^x,uⁿ⁾

n|_L₂₍_Ω₎ ≤c|v_n|, by (3.2) and the Lebesgue dominated convergence theorem, we arrive at Z

R^N×_R^N(v(x)−v(y))(ϕ(x)−ϕ(y))K(x−y)dxdy−

Z

Ωlv+ϕdx=0, for any ϕ∈ X0. (3.3) From the strong maximum principle (see [9]) we deduce thatv>0. Choosing ϕ=φ₁ in (3.3), we obtain

l Z

Ωvφ₁dx= λ₁ Z

Ωvφ₁dx.

This is a contradiction.

Lemma 3.2. Letφ₁be the eigenfunction corresponding toλ₁withkφ₁k=1. If f₀<λ₁ <l, then (a) there existρ,β>0such thatJ₊(u)≥βfor all u∈X₀withkuk=ρ;

(b) J₊(tφ₁) =−_∞as t→+_∞.

Proof. By(f₁)and(f₃), ifl ∈(λ₁,+_∞), for anyε>0, there exist A= A(ε)≥0 andB= B(ε) such that for all(x,s)∈_Ω×_R,

F+(x,s)≤ ¹

2(f₀+ε)s²+As^p⁺¹, (3.4) F+(x,s)≥ ¹

2(l−ε)s²−B, (3.5)

where p∈(_1,^N_N⁺₋^s_s)_.

Chooseε>0 such that f₀+ε<λ₁. By (3.4) and Lemma2.1, we get J+(u) = ¹

2kuk²_X₀ −

Z

ΩF(x,u)dx

≥ ¹ 2kuk²_X

0 −¹

2 Z

Ω[(f₀+ε)u²+A|u|^p⁺¹]dx

= ¹ 2

1− ^f⁰+ε λ₁

kuk²_X₀ −ckuk^p_X⁺¹

0 . So, part(a)holds if we choosekuk_X₀ = ρ>0 small enough.

On the other hand, ifl∈(λ₁,+_∞), takeε>0 such thatl−ε> λ₁. By (3.5), we have J₊(u)≤ ¹

2kuk²_X

0 −^l−ε

2 |u|²_L₂₍_Ω₎+B|_Ω|_. Sincel−ε>λ₁ andkφ₁k_X₀ =1, it is easy to see that

J₊(tφ₁)≤ ¹ 2

1−^l−ε λ₁

t²+B|_Ω| → −_∞ _ast→+_∞ and part(b)_{is proved.}

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Lemma 3.3. Let X₀ = H_k⊕_P_k₊₁. If f satisfies(f₁),(f₃)and(f₄)then (i) the functionalJ is coercive onPk+1, that is

J(u)→+_∞ askuk_X₀ →+_∞, u∈ _P_k₊₁ and bounded from below onPk+1,

(ii) the functional J is anti-coercive on H_k.

Proof. For u ∈ _P_k₊₁, by (f₃), for any ε > 0, there exists B₁ = B₁(ε) such that for all(x,s) ∈ Ω×_R,

F(x,s)≤ ¹

2(l+ε)s²+B₁. (3.6)

So, from Proposition2.3we have J(u) = ¹

2kuk²_X₀−

Z

ΩF(x,u)dx

≥ ¹ 2kuk²_X

0− ¹

2(l+ε)|u|²_L2(_Ω)−B₁|_Ω|

≥ ¹ 2

1− ^l+ε λ_k+₁

kuk²_X

0−B₁|_Ω|. Chooseε >0 such thatl+ε<λ_k+1. This proves(i).

(ii)We firstly consider the casel=λ_k.

WriteG(x,t) =F(x,t)− ¹₂λ_kt²,g(x,t) = f(x,t)−λ_kt. Then(f3)and(f₄)imply that

|tlim|→_∞[g(x,t)t−2G(x,t)] =−_∞ (3.7) and

|tlim|→_∞

2G(x,t)

t² =0. (3.8)

It follows from (3.7) that for everyM >0, there exists a constantT>0 such that

g(x,t)t−2G(x,t)≤ −M, ∀t ∈_R, |t| ≥T, a.e.x∈_Ω. (3.9) For τ>0, we have

d dτ

G(x,τ)

τ² = ^g(x,τ)τ−2G(x,τ)

τ³ . (3.10)

Integrating (3.10) over[t,s]⊂[T,+_∞), we deduce that G(x,s)

s² − ^G(x,t) t² ≤ ^M

2 (¹ s² − ¹

t²). (3.11)

Letting s → +_∞ and using (3.8), we see that G(x,t) ≥ ^M₂, for t ∈ _R, t ≥ T, a.e. x ∈ _{Ω. A} similar argument shows that G(x,t)≥ ^M₂, fort∈_R, t ≤ −T, a.e. x∈_{Ω. Hence}

|tlim|→_∞G(x,t)→+_∞, a.e. x∈_Ω. (3.12)

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By (3.12) and Proposition2.2, we get J(v) = ¹

2kvk²_X

0−

Z

ΩF(x,v)dx

= ¹

2kvk²_X₀− ¹ 2λ_k

Z

Ωv²dx−

Z

ΩG(x,v)dx

≤ −δkv⁻k²_X

0 −

Z

ΩG(x,v)dx→ −_∞ forv ∈Vwithkvk_X₀ →+_{∞, where}v⁻∈ H_k₋₁.

In the case of λ_k < l <λ_k+1, we needn’t the assumption(f₄)and it is easy to see that the conclusion also holds .

Lemma 3.4. Ifλ_k <l<λ_k₊₁, thenJ satisfies the (PS) condition.

Proof. Let{un} ⊂X0 be a sequence such that|J(_u_n)| ≤c,hJ⁰(_u_n)_,ϕi →0. Since hJ⁰(un),ϕi=

Z

R^N×_R^N(un(x)−un(y))(ϕ(x)−ϕ(y))K(x−y)dxdy−

Z

Ω f(x,un)ϕdx

=o(kϕk_X₀)_. _(3.13)

for all ϕ ∈ X₀. If |u_n|_L2(_Ω) is bounded, we can take ϕ = u_n. By (f₃), there exists a constant c>0 such that|f(x,un(x))| ≤c|un(x)|, a.e.x ∈_{Ω. So}unis bounded inX₀. If|un|_L2(_Ω)→+_∞, asn→_{∞, set}v_n= _| ^uⁿ

un|_L₂₍_Ω₎, then|v_n|_L2(_Ω)=1. Taking ϕ= v_nin (3.13), it follows thatkv_nk_X₀ is bounded. Without loss of generality, we assumev_n *v inX₀, thenv_n→ vin L²(_Ω). Hence, v_n →va.e. inΩ. Dividing both sides of (3.13) by|u_n|_L2(_Ω), we get

Z

R^N×_R^N

(v_n(x)−v_n(y))(ϕ(x)−ϕ(y))K(x−y)dxdy−

Z

Ω

f(x,u_n)

|un|_L2(_Ω)

ϕdx

=o kϕk_X₀

|u_n|_L2(_Ω)

!

, ∀ϕ∈X₀. (3.14)

Then for a.e. x∈ _{Ω, we have} _|_u^f⁽^x,uⁿ⁾

n|_L₂₍_Ω₎ →lvasn→∞. In fact, ifv(x)6=0, by(f3), we have

|un(x)|=|vn(x)||un|_L2(_Ω)→+_∞

and f(x,u_n)

|u_n|_L2(_Ω)

= ^f(x,u_n)

u_n v_n→lv.

Ifv(x) =0, we have

|f(x,u_n)|

|un|_L2(_Ω)

≤c|vn| −→0.

Since ^|_|_u^f⁽^x,uⁿ^)|

n|_L₂₍_Ω₎ ≤ c|v_n|, by (3.14) and the Lebesgue dominated convergence theorem, we arrive at

Z

Ωlvϕdx=0, for any ϕ∈ X₀. Obviouslyv6=0, hence,lis an eigenvalue of(−_∆)^s. This contradicts our assumption.

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Lemma 3.5. Suppose l=λ_k and f satisfies(f₄). Then the functionalJ satisfies the (C) condition.

Proof. Supposeu_n∈ X₀satisfies

J(u_n)→c∈_R, (1+ku_nk)kJ⁰(u_n)k →0 as n→_∞. (3.15) In view of(f₃), it suffices to prove thatunis bounded inX₀. Similar to the proof of Lemma3.4, we have

Z

Ωlvϕdx=0, for any ϕ∈ X₀. (3.16) Therefore v 6= 0 is an eigenfunction of λ_k, then |un(x)| → _∞for a.e. x ∈ _Ω₀ _(Ω₀ ⊂ _{Ω) with} positive measure. It follows from(f₄)that

n→+lim_∞[f(x,un(x))un(x)−2F(x,un(x))] =−_∞ holds uniformly inx∈ _Ω₀, which implies that

Z

Ω(f(x,u_n)u_n−2F(x,u_n))dx→ −_∞ asn→_∞. (3.17) On the other hand, (3.15) implies that

2J(u_n)− hJ⁰(u_n),u_ni →2c asn→_∞.

Thus _Z

Ω(f(x,u_n)u_n−2F(x,u_n))dx→2c as n→_∞, which contradicts (3.17). Henceu_nis bounded.

It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [6] for more information on Morse theory.

Let X be a Hilbert space and J ∈ C¹(X,R)be a functional satisfying the (PS) condition or (C) condition, and Hq(X,Y) be the q-th singular relative homology group with integer coefficients. Let u₀ be an isolated critical point of J with J(u₀) = c, c ∈ _{R, and} U be a neighborhood of u₀. The group

C_q(J,u₀):= H_q(J^c∩U,J^c∩U\{u₀}), q∈Z

is said to be the q-th critical group ofJ atu₀, where J^c={u∈ X:J(u)≤c}.

Let K := {u ∈ X : J⁰(u) = 0} be the set of critical points of J and a < infJ(K), the critical groups ofJ at infinity are formally defined by (see [3])

C_q(J_,_∞)_:= H_q(X,J^a)_, q∈Z.

The following result comes from [3,6] and will be used to prove the results in this paper.

Proposition 3.6([3]). Assume that X=V⊕W,J is bounded from below on W andJ(u)→ −_∞ askuk →_∞with u∈V. Then

C_k(J,∞)^0, ^{if k}=dimV <_∞. (3.18)

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4 Proof of the main results

Proof of Theorem1.1. By Lemmas 3.1, 3.2 and the mountain pass theorem, the functionalJ+

has a critical pointu₁ satisfying J₊(u₁)≥ β. SinceJ₊(0) =0, u₁ 6= 0 and by the maximum principle (see [9]), we get u₁ > 0. Hence u₁ is a positive solution of the problem (1.1) and satisfies

C₁(J₊,u₁)6=0, u₁>0. (4.1) By(f₂), the functionalJ isC². Using the results in [6,10], we obtain

C_q(J,u₁) =C_q(J_C0

d(_Ω),u₁) =C_q(J₊|_C0

d(_Ω),u₁) =C_q(J₊,u₁) =δ_q1Z. (4.2) Here

C_d⁰(_Ω) ={u∈C⁰(_Ω):ud⁻^γ ∈C⁰(_Ω)},

where d(x) = _dist(x,∂Ω) _{for all} x ∈ _Ω _{and 0}< γ < 1. More detailed topology knowledge will be seen in [9] and we omit it.

Similarly, we can obtain another negative critical pointu₂ ofJ satisfying

Cq(J,u2) =δ_q,1Z. (4.3)

Since f0< λ₁, the zero function is a local minimizer ofJ, then

C_q(J, 0) =δ_q,0Z. (4.4)

On the other hand, by Lemmas3.3,3.4 and Proposition3.6, we have

C_k(J,∞)^0. ^(4.5)

HenceJ has a critical point u₃satisfying

C_k(J,u₃)^0. ^(4.6)

Sincek≥2, it follows from (4.2)–(4.6) thatu₁,u₂andu₃are three different nontrivial solutions of the problem (1.1).

Proof of Theorem1.2. By Lemmas 3.3, 3.5 and Proposition 3.6, we can prove the conclusion (4.5). The other proof is similar to that of Theorem1.1.

Acknowledgments

This research was supported by the NSFC (Nos. 11571176 and 11561059), NSF of Gansu Province (No. 1506RJZE114), Planned Projects for Postdoctoral Research Funds of Jiangsu Province (No. 1301038C), TSNC (No. TSA1406) and Scientific Research Foundation of the Higher Education Institutions of Gansu Province (No. 2015A-131). The authors would like to thank the anonymous referees for useful suggestions.

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