Existence results for a class of nonlocal problems involving p-Laplacian

Abstract

This paper is concerned with the existence of solutions to a class of p-Kirchhoff type equations with Neumann boundary data as follows:

$$\left\{\begin{array}{cc}\hfill -{\left[M\left({{\int }_{\Omega }\left|\nabla u\right|}^{p}dx\right)\right]}^{p-1}{\Delta }_{p}u=f\left(x,u\right),\hfill & \hfill \text{in}\Omega ;\hfill \\ \hfill \frac{\partial u}{\partial \upsilon }=0,\hfill & \hfill \text{on}\partial \Omega .\hfill \end{array}\right.$$

By means of a direct variational approach, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

1. Introduction

In this paper, we deal with the nonlocal p-Kirchhoff type of problem given by:

$$\left\{\begin{array}{cc}\hfill -{\left[M\left({\int }_{\Omega }{\left|\nabla u\right|}^{p}dx\right)\right]}^{p-1}{\Delta }_{p}u=f\left(x,u\right),\hfill & \hfill \text{in}\Omega ;\hfill \\ \hfill \frac{\partial u}{\partial \upsilon }=0,\hfill & \hfill \text{on}\partial \Omega \hfill \end{array}\right.$$

(1.1)

where Ω is a smooth bounded domain in R^N, 1 < p < N, ν is the unit exterior vector on ∂Ω, Δ _p is the p-Laplacian operator, that is, Δ _p u = div(|∇u|^p−2∇u), the function M : R⁺ → R⁺ is a continuous function and there is a constant m₀ > 0, such that

$$\left(M_0\right)M\left(t\right)\ge m_0\text{for all}t\ge 0.$$

$f\left(x,t\right):\overline{\Omega }\times \text{R}\to \text{R}$ is a continuous function and satisfies the subcritical condition:

$$\left|f\left(x,t\right)\right|\le C\left({\left|t\right|}^{q-1}+1\right),\text{for some}p<q<p^{*}=\left\{\begin{array}{cc}\hfill \frac{Np}{N-p},\hfill & \hfill N\ge 3;\hfill \\ \hfill +\infty ,\hfill & \hfill N=1,2.\hfill \end{array}\right.$$

(1.2)

where C denotes a generic positive constant.

Problem (1.1) is called nonlocal because of the presence of the term M, which implies that the equation is no longer a pointwise identity. This provokes some mathematical difficulties which makes the study of such a problem particulary interesting. This problem has a physical motivation when p = 2. In this case, the operator M(∫_Ω|∇u|²dx)Δu appears in the Kirchhoff equation which arises in nonlinear vibrations, namely

$$\left\{\begin{array}{cc}\hfill u_{tt}-M\left({\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\right)\Delta u=f\left(x,u\right),\hfill & \hfill \text{in}\Omega \times \left(0,T\right);\hfill \\ \hfill u=0,\hfill & \hfill \text{on}\partial \Omega \times \left(0,T\right);\hfill \\ \hfill u\left(x,0\right)=u_0\left(x\right),\hfill & \hfill u_t\left(x,0\right)=u_1\left(x\right).\hfill \end{array}\right.$$

P-Kirchhoff problem began to attract the attention of several researchers mainly after the work of Lions [1], where a functional analysis approach was proposed to attack it. The reader may consult [2–8] and the references therein for similar problem in several cases.

This work is organized as follows, in Section 2, we present some preliminary results and in Section 3 we prove the main results.

2. Preliminaries

By a weak solution of (1.1), then we say that a function u ε W^1,p(Ω) such that

$${\left[M\left({\int }_{\Omega }{\left|\nabla u\right|}^{p}dx\right)\right]}^{p-1}{\int }_{\Omega }{\left|\nabla u\right|}^{p-2}\nabla u\nabla \phi dx={\int }_{\Omega }f\left(x,u\right)\phi dx,\text{for all}\phi \in W^{1,p}\left(\Omega \right)$$

So we work essentially in the space W^1,p(Ω) endowed with the norm

$$∥u∥={\left({\int }_{\Omega }\left({\left|\nabla u\right|}^p+{\left|u\right|}^p\right)dx\right)}^{\frac{1}{p}},$$

and the space W^1,p(Ω) may be split in the following way. Let W _c = 〈1〉, that is, the subspace of W^1,p(Ω) spanned by the constant function 1, and $W_0=\left\{z\in W^{1,p}\left(\Omega \right),{\int }_{\Omega }z=0\right\}$, which is called the space of functions of W^1,p(Ω) with null mean in Ω. Thus

$$W^{1,p}\left(\Omega \right)=W_0\oplus W_c.$$

As it is well known the Poincaré's inequality does not hold in the space W^1,p(Ω). However, it is true in W₀.

Lemma 2.1[8] (Poincaré-Wirtinger's inequality) There exists a constant η > 0 such that${\int }_{\Omega }{\left|z\right|}^{p}dx\le \eta {\int }_{\Omega }{\left|\nabla z\right|}^{p}dx$for all z ∈ W₀.

Let us also recall the following useful notion from nonlinear operator theory. If X is a Banach space and A : X → X* is an operator, we say that A is of type (S₊), if for every sequence {x _n }_n≥1⊆ X such that x _n ⇀ x weakly in X, and $lim{sup}_{n\to \infty }⟨A\left(x_n\right),x_n-x⟩\le 0$. we have that x _n → x in X.

Let us consider the map A : W^1,p(Ω) → W^1,p(Ω)* corresponding to −Δ _p with Neumann boundary data, defined by

$$⟨A\left(u\right),v⟩={\int }_{\Omega }{\left|\nabla u\right|}^{p-2}\nabla u\nabla vdx,\forall u,v\in W^{1,p}\left(\Omega \right).$$

(2.1)

We have the following result:

Lemma 2.2[9, 10]The map A : W^1,p(Ω) → W^1,p(Ω)* defined by (2.1) is continuous and of type (S₊).

In the next section, we need the following definition and the lemmas.

Definition 2.1. Let E be a real Banach space, and D an open subset of E. Suppose that a functional J : D → R is Fréchet differentiable on D. If x₀ ∈ D and the Fréchet derivative J' (x₀) = 0, then we call that x₀is a critical point of the functional J and c = J(x₀) is a critical value of J.

Definition 2.2. For J ∈ C¹(E, R), we say J satisfies the Palais-Smale condition (denoted by (PS)) if any sequence {u _n } ⊂ E for which J(u _n ) is bounded and J'(u _n ) → 0 as n → ∞ possesses a convergent subsequence.

Lemma 2.3[11]Let X be a Banach space with a direct sum decomposition X = X₁ ⊕ X₂, with k = dimX₂ < ∞, let J be a C¹function on X, satisfying (PS) condition. Assume that, for some r > 0,

$$\begin{array}{c}J\left(u\right)\le 0foru\in X_1,∥u∥\le r;\\ J\left(u\right)\ge 0foru\in X_2,∥u∥\le r.\end{array}$$

Assume also that J is bounded below and inf _X J < 0. Then J has at least two nonzero critical points.

Lemma 2.4[12]Let X = X₁ ⊕ X₂, where X is a real Banach space and X₂ ≠ {0}, and is finite dimensional. Suppose J ∈ C¹(X, R) satisfies (PS) and

(i) there is a constant α and a bounded neighborhood D of 0 in X₂such that J| _∂D ≤ α and,

(ii) there is a constant β > α such that${J{\mid }_X}_{{}_1}\ge \beta$,

then J possesses a critical value c ≥ β, moreover, c can be characterized as

$$c=\underset{h\in \Gamma }{inf}\underset{u\in \overline{D}}{max}J\left(h\left(u\right)\right).$$

where$\Gamma =\left\{h\in C\left(\overline{D},X\right)\mid h=idon\partial D\right\}$.

Definition 2.3. For J ∈ C¹(E, R), we say J satisfies the Cerami condition (denoted by (C)) if any sequence {u _n } ⊂ E for which J(u _n ) is bounded and (1 ||u _n ||) J'(u _n )|| → 0 as n → ∞ possesses a convergent subsequence.

Remark 2.1 If J satisfies the (C) condition, Lemma 2.4 still holds.

In the present paper, we give an existence theorem and a multiplicity theorem for problem (1.1). Our main results are the following two theorems.

Theorem 2.1 If following hold:

(F₀) $0\le {lim}_{\mid u\mid \to 0}\frac{pF\left(x,u\right)}{{\left|u\right|}^p}<\frac{m_0^{p-1}}{\eta }a.e.x\in \Omega$, where$F\left(x,u\right)={\int }_0^{u}f\left(x,s\right)ds$, η appears in Lemma 2.1 ;

(F₁) ${lim}_{\mid u\mid \to \infty }\frac{pF\left(x,u\right)}{{\left|u\right|}^p}\le 0a.e.x\in \Omega$;

(F₂)${lim}_{\mid u\mid \to \infty }{\int }_{\Omega }F\left(x,u\right)dx=-\infty$.

Then the problem (1.1) has least three distinct weak solutions in W^1,p(Ω).

Theorem 2.2 If the following hold:

(M₁) The function M that appears in the classical Kirchhoff equation satisfies$\hat{M}\left(t\right)\le {\left(M\left(t\right)\right)}^{p-1}t$for all t ≥ 0, where$\hat{M}\left(t\right)={\int }_0^t{\left[M\left(s\right)\right]}^{p-1}ds$;

(F₃)$f\left(x,u\right)u>0forallu\ne 0$;

(F₄)${lim}_{{}_{\mid u\mid \to \infty }}\frac{pF\left(x,u\right)}{{\left|u\right|}^p}=0a.e.x\in \Omega$;

(F₅)${lim}_{\mid u\mid \to \infty }\left(f\left(x,u\right)u-pF\left(x,u\right)\right)=-\infty$.

Then the problem (1.1) has at least one weak solution in W^1,p(Ω).

Remark 2.2 We exhibit now two examples of nonlinearities that fulfill all of our hypotheses

$$f\left(x,u\right)=\frac{m_0^{p-1}}{2\eta }{\left|u\right|}^{p-2}u-{\left|u\right|}^{q-2}u,$$

hypotheses (F₀), (F₁), (F₂) and (1.2) are clearly satisfied.

$$f\left(x,u\right)=arctanu+\frac{u}{1+u^2},$$

hypotheses (F₃), (F₄) and (F₅) and (1.2) are clearly satisfied.

3. Proofs of the theorems

Let us start by considering the functional J : W^1,p(Ω) → R given by

$$J\left(u\right)=\frac{1}{p}\hat{M}\left({\int }_{\Omega }{\left|\nabla u\right|}^{p}dx\right)-{\int }_{\Omega }F\left(x,u\right)dx.$$

Proof of Theorem 2.1 By (F₀), we know that f(x, 0) = 0, and hence u(x) = 0 is a solution of (1.1).

To complete the proof we prove the following lemmas.

Lemma 3.1 Any bounded (PS) sequence of J has a strongly convergent subsequence.

Proof: Let {u _n } be a bounded (PS) sequence of J. Passing to a subsequence if necessary, there exists u ∈ W^1,p(Ω) such that u _n ⇀ u. From the subcritical growth of f and the Sobolev embedding, we see that

$${\int }_{\Omega }f\left(x,u_n\right)\left(u_n-u\right)dx\to 0.$$

and since J'(u _n )(u _n − u) → 0, we conclude that

$${\left[M\left({{\int }_{\Omega }\left|\nabla u_n\right|}^{p}dx\right)\right]}^{p-1}{\int }_{\Omega }{\left|\nabla u_n\right|}^{p-2}\nabla u_n\nabla \left(u_n-u\right)dx\to 0.$$

In view of condition (M₀), we have

$${\int }_{\Omega }{\left|\nabla u_n\right|}^{p-2}\nabla u_n\nabla \left(u_n-u\right)dx\to 0.$$

Using Lemma 2.2, we have u _n → u as n → ∞. □

Lemma 3.2 If condition (M₀), (F₁) and (F₂) hold, then${lim}_{\parallel u\parallel \to \infty }J\left(u\right)=+\infty$.

Proof: If there are a sequence {u _n } and a constant C such that ||u _n || → ∞ as n → ∞, and J(u _n ) ≤ C (n = 1, 2 ···), let $v_n=\frac{u_n}{∥u_n∥}$, then there exist v₀ ∈ W^1,p(Ω) and a subsequence of {v _n }, we still note by {v _n }, such that v _n ⇀ v₀ in W^1,p(Ω) and v _n → v₀ in L^p (Ω).

For any ε > 0, by (F₁), there is a H > 0 such that $F\left(x,u\right)\le \frac{\epsilon }{p}{\left|u\right|}^p$ for all |u| ≥ H and a.e. x ∈ Ω, then there exists a constant C > 0 such that $F\left(x,u\right)\le \frac{\epsilon }{p}{\left|u\right|}^p+C$ for all u ∈ R, and a.e. x ∈ Ω, Consequently

$$\begin{array}{ll}\hfill \frac{C}{\parallel u_n\mid {\mid }^p}& \ge \frac{J\left(u_n\right)}{\parallel u_n\mid {\mid }^p}=\frac{1}{\parallel u_n\mid {\mid }^p}\left(\frac{1}{p}\hat{M}\left({\int }_{\Omega }\mid \nabla u_n{\mid }^{p}dx\right)-{\int }_{\Omega }F\left(x,u_n\right)dx\right)\\ \ge \frac{1}{p}{m}_0^{p-1}{\int }_{\Omega }\mid \nabla v_n{\mid }^{p}dx-\frac{\epsilon }{p}{\int }_{\Omega }\mid v_n{\mid }^{p}dx-\frac{C\mid \Omega \mid }{\parallel u_n\mid {\mid }^p}\\ =\frac{1}{p}{m}_0^{p-1}-\left(\frac{1}{p}{m}_0^{p-1}+\frac{\epsilon }{p}\right){\int }_{\Omega }\mid v_n{\mid }^{p}dx-\frac{C\mid \Omega \mid }{\parallel u_n\mid {\mid }^p}.\end{array}$$

It implies ∫_Ω|v₀| ^p dx ≥ 1. On the other hand, by the weak lower semi-continuity of the norm, one has

$$\parallel v_0\parallel \le \underset{n\to \infty }{lim inf}\parallel v_n\parallel =1.$$

Hence ${\int }_{\Omega }\mid \nabla v_0{\mid }^{p}dx=0$, so |v₀(x)| = constant ≠ 0 a.e. x ∈ Ω. By (F₂), ${\mathsf{\text{lim}}}_{\mid u_n\mid \to \infty }{\int }_{\Omega }F\left(x,u_n\right)\mathsf{\text{d}}x\to -\infty$. Hence

$$\begin{array}{ll}\hfill C\ge J\left(u_n\right)& =\frac{1}{p}\hat{M}\left({\int }_{\Omega }\mid \nabla u_n{\mid }^{p}dx\right)-{\int }_{\Omega }F\left(x,u_n\right)dx\\ \ge -{\int }_{\Omega }F\left(x,u_n\right)dx\to +\infty asn\to \infty .\end{array}$$

This is a contradiction. Hence J is coercive on W^1,p(Ω), bounded from below, and satisfies the (PS) condition. □

By Lemma 3.1 and 3.2, we know that J is coercive on W^1,p(Ω), bounded from below, and satisfies the (PS) condition. From condition (F₀), we know, there exist r > 0, ε > 0 such that

$$0\le F\left(x,u\right)\le \left(\frac{m_0^{p-1}}{p\eta }-\epsilon \right)\mid u{\mid }^p,for\mid u\mid \le r.$$

If u ∈ W _c , for ||u|| ≤ ρ₁, then |u| ≤ r, we have

$$\begin{array}{ll}\hfill J\left(u\right)& =\frac{1}{p}\hat{M}\left({\int }_{\Omega }\mid \nabla u{\mid }^{p}dx\right)-{\int }_{\Omega }F\left(x,u\right)dx\\ =-{\int }_{\Omega }F\left(x,u\right)dx\le 0.\end{array}$$

If u ∈ W₀, then from condition (F₀) and (1.2), we have

$$F\left(x,u\right)\le \left(\frac{m_0^{p-1}}{p\eta }-\epsilon \right)\mid u{\mid }^p+C\mid u{\mid }^q,foru\in R,q\in \left(p,p^{*}\right).$$

Noting that

$${\int }_{\Omega }\mid u{\mid }^{p}dx\le \eta {\int }_{\Omega }\mid \nabla u{\mid }^{p}dx,u\in W_0,$$

we can obtain

$$\begin{array}{ll}\hfill J\left(u\right)& =\frac{1}{p}\hat{M}\left({\int }_{\Omega }\mid \nabla u{\mid }^{p}dx\right)-{\int }_{\Omega }F\left(x,u\right)dx\\ \ge \frac{1}{p}{m}_0^{p-1}{\int }_{\Omega }\mid \nabla u{\mid }^{p}dx-\frac{m_0^{p-1}}{p\eta }{\int }_{\Omega }\mid u{\mid }^{p}dx+\epsilon {\int }_{\Omega }\mid u{\mid }^{p}dx-C{\int }_{\Omega }\mid u{\mid }^{q}dx\\ \ge C\epsilon \parallel u\mid {\mid }^p-C{C}_1\parallel u\mid {\mid }^q.\end{array}$$

Choose ||u|| = ρ₂ small enough, such that J(u) ≥ 0 for ||u|| ≤ ρ₂ and u ∈ W₀.

Now choose ρ = min{ρ₁, ρ₂}, then, we have

$$J\left(u\right)\le 0foru\in W_c,\parallel u\parallel \le \rho ;$$

$$J\left(u\right)\le 0foru\in W_0,\parallel u\parallel \le \rho .$$

If inf{J(u), u ∈ W^1,p(Ω)} = 0, then all u ∈ W _c with ||u|| ≤ ρ are minimum of J, which implies that J has infinite critical points. If inf{J(u), u ∈ W^1,p(Ω)} < 0 then by Lemma 2.3, J has at least two nontrivial critical points. Hence problem (1.1) has at least two nontrivial solutions in W^1,p(Ω), Therefore, problem (1.1) has at least three distinct solutions in W^1,p(Ω). □

Proof of Theorem 2.2. We divide the proof into several lemmas.

Lemma 3.3 If condition (F₃) and (F₅) hold, then$J{\mid }_{W_c}$is anticoercive. (i.e. we have that J(u) → -∞, as |u| → ∞, u ∈ R.)

Proof: By virtue of hypothesis (F₅), for any given L > 0, we can find R₁ = R₁(L) > 0 such that

$$F\left(x,u\right)\ge \frac{1}{p}L+\frac{1}{p}f\left(x,u\right)u,fora.e.x\in \Omega ,\mid u\mid >R_1.$$

Thus, using hypothesis (F₃), we have

$$F\left(x,u\right)\ge \frac{1}{p}L-C,fora.e.x\in \Omega u\in \text{R}$$

So

$${\int }_{\Omega }F\left(x,u\right)dx\ge \frac{1}{p}L\mid \Omega \mid -C\mid \Omega \mid .$$

Since L > 0 is arbitrary, it follows that

$${\int }_{\Omega }F\left(x,u\right)dx\to \infty ,as\mid u\mid \to \infty ,$$

and so

$$J\left(u\right){\mid }_{W_C}=-{\int }_{\Omega }F\left(x,u\right)dx\to -\infty ,as\mid u\mid \to \infty .$$

This proves that $J{\mid }_{W_c}$ is anticoercive. □

Lemma 3.4 If hypothesis (F₄) holds, then$J{\mid }_{W_0}\ge -\infty$.

Proof: For a given $0<\epsilon <m_0^{p-1}$, we can find C _ε > 0 such that $F\left(x,u\right)\le \frac{\epsilon }{p\eta }\mid u{\mid }^p+C_{\epsilon }$ for a.e. x ∈ Ω all u ∈ R. Then

$$\begin{array}{ll}\hfill J\left(u\right){\mid }_{u\in W_0}& =\frac{1}{p}\hat{M}\left({\int }_{\Omega }\mid \nabla u{\mid }^{p}dx\right)-{\int }_{\Omega }F\left(x,u\right)dx\\ \ge \frac{1}{p}{m}_0^{p-1}{\int }_{\Omega }\mid \nabla u{\mid }^{p}dx-\frac{m_0^{p-1}}{p\eta }{\int }_{\Omega }\mid u{\mid }^{p}dx-C\mid \Omega \mid \\ \ge -C\mid \Omega \mid .\end{array}$$

then $J{\mid }_{W_0}\ge -\infty$. □

Lemma 3.5 If condition (F₄) (F₅) hold, then J satisfies the (C) condition.

Proof: Let {u _n }_{n ≥1}⊆ W^1,p(Ω) be a sequence such that

$$\mid J\left(u_n\right)\mid \le M_1,\forall n\ge 1.$$

(3.1)

with some M₁ > 0 and

$$\left(1+\parallel u_n\parallel \right)J^{\prime }\left(u_n\right)\to 0,\text{in}{W}^{1,p}{\left(\Omega \right)}^{*}asn\to \infty .$$

(3.2)

We claim that the sequence {u _n } is bounded. We argue by contradiction. Suppose that ||u|| → +∞, as n → ∞, we set $v_n=\frac{u_n}{∥u_n∥}$, ∀n ≥ 1. Then ||v _n || = 1 for all n ≥ 1 and so, passing to a subsequence if necessary, we may assume that

$$v_n\rightharpoonup v\text{in}{W}^{1,p}\left(\Omega \right);$$

$$v_n\to v\text{in}{L}^p\left(\Omega \right).$$

from (3.2), we have ∀h ∈ W^1,p(Ω)

$$\left|{\left[M\left({\int }_{\Omega }\mid \nabla u_n{\mid }^{p}dx\right)\right]}^{p-1}{\int }_{\Omega }\mid \nabla v_n{\mid }^{p-2}\nabla v_n\nabla hdx-{\int }_{\Omega }\frac{f\left(x,u_n\right)h}{{∥u_n∥}^{p-1}}dx\right|\le \frac{{\epsilon }_n}{1+∥u_n∥}\frac{∥h∥}{{∥u_n∥}^{p-1}}$$

(3.3)

with ε _n ↓ 0.

In (3.3), we choose h = v _n − v ∈ W^1,p(Ω), note that by virtue of hypothesis (F₄), we have

$$\frac{f\left(x,u_n\right)}{\parallel u_n\mid {\mid }^{p-1}}\rightharpoonup 0\text{in}{L}^{p^{\prime }}\left(\Omega \right),$$

where $\frac{1}{p}+\frac{1}{p^{\prime }}=1$.

So we have

$${\left[M\left({\int }_{\Omega }\mid \nabla u_n{\mid }^{p}dx\right)\right]}^{p-1}{\int }_{\Omega }\mid \nabla v_n{\mid }^{p-2}\nabla v_n\nabla \left(v_n-v\right)dx\to 0.$$

Since M(t) > m₀ for all t ≥ 0, so we have

$${\int }_{\Omega }\mid \nabla v_n{\mid }^{p-2}\nabla v_n\nabla \left(v_n-v\right)dx\to 0.$$

Hence, using the (S₊) property, we have v _n → v in W^1,p(Ω) with ||v|| = 1, then v ≠ 0. Now passing to the limit as n → ∞ in (3.3), we obtain

$${\int }_{\Omega }\mid \nabla v{\mid }^{p-2}\nabla v\nabla hdx\to 0,\forall h\in W^{1,p}\left(\Omega \right),$$

then v = ξ ∈ R. Then |u _n (x)| → +∞ as n → +∞. Using hypothesis (F₅), we have f(x, u _n (x))u _n (x) - pF(x, u _n (x)) → -∞ for a.e x ∈ Ω.

Hence by virtue of Fatou's Lemma, we have

$${\int }_{\Omega }f\left(x,u_n\right)u_n-pF\left(x,u_n\right)dx\to -\infty ,asn\to +\infty .$$

(3.4)

From (3.1), we have

$$\hat{M}\left({\int }_{\Omega }{\left|\nabla u_n\mid \right.}^p\right)dx-p{\int }_{\Omega }F\left(x,u_n\right)dx\ge -p{M}_1,\forall n\ge 1.$$

(3.5)

From (3.2), we have

$$\left|{\left[M\left({\int }_{\Omega }\mid \nabla u_n{\mid }^{p}dx\right)\right]}^{p-1}{\int }_{\Omega }{\left|\nabla u_n\right|}^{p-2}\nabla u_n\nabla hdx-{\int }_{\Omega }f\left(x,u_n\right)hdx\right|\le \frac{{\epsilon }_n\parallel h\parallel }{1+\parallel u_n\parallel }\forall h\in W^{1,p}\left(\Omega \right).$$

With ε _n ↓ 0. So choosing h = u _n ∈ W^1,p(Ω), we obtain

$$-{\left[M\left({\int }_{\Omega }{\left|\nabla u_n\right|}^{p}dx\right)\right]}^{p-1}{\int }_{\Omega }{\left|\nabla u_n\right|}^{p}dx+{\int }_{\Omega }f\left(x,u_n\right)u_{n}dx\ge -{\epsilon }_n.$$

(3.6)

Adding (3.5) and (3.6), noting that $\hat{M}\left(t\right)\le {\left(M\left(t\right)\right)}^{p-1}t$ for all t ≥ 0, we obtain

$${\int }_{\Omega }\left(f\left(x,u_n\right)u_n-pF\left(x,u_n\right)\right)dx\ge -M_2,\forall n\ge 1,$$

(3.7)

comparing (3.4) and (3.7), we reach a contradiction. So {u _n }in bounded in W^1,p(Ω). Similar with the proof of Lemma 3.1, we know that J satisfied the (C) condition. □

Sum up the above fact, from Lemma 2.4 and Remark 2.1, Theorem 2.2 follows from the Lemma 3.3 to 3.5.