Multiple solutions for the $p(x)$-Laplacian problem involving critical growth with aparameter

Abstract

By energy estimates and establishing a local ${(\mathit{PS})}_c$ condition, existence of solutions for the$p(x)$-Laplacian problem involving critical growth in abounded domain is obtained via the variational method under the presence ofsymmetry.

MSC: 35J20, 35J62.

1 Introduction

In recent years, the study of problems in differential equations involving variableexponents has been a topic of interest. This is due to their applications in imagerestoration, mathematical biology, dielectric breakdown, electrical resistivity,polycrystal plasticity, the growth of heterogeneous sand piles and fluid dynamics,etc. We refer readers to [1–7] for more information. Furthermore, new applications are continuing to appear,see, for example, [8] and the references therein.

With the variational techniques, the $p(x)$-Laplacian problems with subcritical nonlinearities havebeen investigated, see [9–13]etc. However, the existence of solutions for $p(x)$-Laplacian problems with critical growth is relatively new.In 2010, Bonder and Silva [14] extended the concentration-compactness principle of Lions to the variableexponent spaces, and a similar result can be found in [15]. After that, there have been many publications for this case, see [16–19]etc.

In this paper, we study the existence and multiplicity of solutions for the quasilinearelliptic problem

$$\{\begin{array}{cc}-{\mathrm{\Delta }}_{p(x)}u=\lambda {|u|}^{q(x)-2}u+f(x,u),\hfill & x\in \mathrm{\Omega };\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$$

(1.1)

where ${\mathrm{\Delta }}_{p(x)}u=div({|\mathrm{\nabla }u|}^{p(x)-2}\mathrm{\nabla }u)$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) is a bounded domain with smooth boundary,$\lambda >0$ is a real parameter, $p(x)$, $q(x)$ are continuous functions on $\overline{\mathrm{\Omega }}$ with

$$1<p^{-}:=\underset{x\in \overline{\mathrm{\Omega }}}{min}p(x)\le p^{+}:=\underset{x\in \overline{\mathrm{\Omega }}}{max}p(x)<N,1\le q(x)\le p^{\ast }(x),\mathrm{\forall }x\in \overline{\mathrm{\Omega }},$$

(1.2)

where

$$p^{\ast }(x)=\frac{Np(x)}{N-p(x)},\mathrm{\forall }x\in \overline{\mathrm{\Omega }},$$

and

$$\{x\in \overline{\mathrm{\Omega }},q(x)=p^{\ast }(x)\}\ne \mathrm{\varnothing }.$$

(1.3)

Related to f, we assume that $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying$sup\{|f(x,s)|;x\in \mathrm{\Omega },|s|\le M\}<\mathrm{\infty }$ for every $M>0$, and the subcritical growth condition:

(f₁) $f(x,s)\le C_1(1+{|s|}^{\beta (x)-1})$ for all $(x,s)\in \mathrm{\Omega }\times \mathbb{R}$, where $\beta (x)$ is a continuous function in $\overline{\mathrm{\Omega }}$ satisfying $\beta (x)<p^{\ast }(x)$, $\mathrm{\forall }x\in \overline{\mathrm{\Omega }}$.

For $F(x,s)={\int }_0^{s}f(x,t)dt$, we suppose that f satisfies the following:

(f₂) there are constants $\sigma \in [0,p^{-})$ and $a_1,a_2>0$ such that for every $s\in \mathbb{R}$, a.e. in Ω,

$$\frac{1}{p^{+}}f(x,s)s-F(x,s)\ge -a_1-a_2{|s|}^{\sigma };$$

(f₃) there are constants $b_1,b_2>0$ and a continuous function $r(x)<p^{\ast }(x)$, $\mathrm{\forall }x\in \overline{\mathrm{\Omega }}$, with $r^{+}>p^{-}$, such that for every $s\in \mathbb{R}$, a.e. in Ω,

$$F(x,s)\le b_1{|s|}^{r(x)}+b_2;$$

(f₄) there are $c_1>0$, $h_1\in L^{p^{\prime }(x)}(\mathrm{\Omega })$ and ${\mathrm{\Omega }}_0\subset \mathrm{\Omega }$ with $|{\mathrm{\Omega }}_0|>0$ such that

$$F(x,s)\ge -h_1(x){|s|}^{p(x)}-c_1\text{for every }s\in \mathbb{R},\text{ a.e. in }\mathrm{\Omega },$$

and

$$\underset{|s|\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{F(x,s)}{{|s|}^{p^{+}}}=\mathrm{\infty }\text{uniformly a.e. in }{\mathrm{\Omega }}_0.$$

Now we state our result.

Theorem 1.1 Assume that (1.2), (1.3) and(f₁)-(f₄) are satisfied with$p^{+}<q^{-}$, $f(x,s)$is odd in s. Then,given$k\in \mathbb{N}$, there exists${\lambda }_k\in (0,\mathrm{\infty }]$such that problem (1.1) possesses atleast k pairs of nontrivial solutions forall$\lambda \in (0,{\lambda }_k)$.

Our paper is motivated by [17]. In [17], the authors considered the multiple solutions to problem (1.1) under theconditions that f has the form $f(x,t)=a(x){|t|}^{p(x)-2}t+g(x,t)$ with $a\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ and g satisfies the following:

(g₁) there is $\alpha >0$ such that

$${\int }_{\mathrm{\Omega }}\frac{1}{p(x)}({|\mathrm{\nabla }u|}^{p(x)}-a(x){|u|}^{p(x)})dx\ge \alpha {\int }_{\mathrm{\Omega }}\frac{1}{p(x)}{|u|}^{p(x)}dx;$$

(g₂) $g\in C(\overline{\mathrm{\Omega }}\times \mathbb{R},\mathbb{R})$, odd with respect to t and

$$\begin{array}{c}g(x,t)=o\left({|t|}^{p(x)-1}\right),|t|\to 0\text{ uniformly in }x,\hfill \\ g(x,t)=o\left({|t|}^{q(x)-1}\right),|t|\to \mathrm{\infty }\text{ uniformly in }x;\hfill \end{array}$$

(g₃) $G(x,t)\le \frac{1}{p^{+}}g(x,t)t$ for all $t\in \mathbb{R}$ and a.e. in Ω, where $G(x,t)={\int }_0^{t}g(x,s)ds$.

Moreover, they assumed that

$$p(x)=p^{+},\mathrm{\forall }x\in \mathrm{\Gamma }=\{x\in \mathrm{\Omega }:a(x)>0\},$$

(1.4)

and the result is the following theorem.

Theorem 1.2 Assume that (1.2), (1.3), (1.4) and(g₁)-(g₃) are satisfied with$p^{+}<q^{-}$. Then there exists asequence$\{{\lambda }_k\}\subset (0,\mathrm{\infty })$with${\lambda }_k>{\lambda }_{k+1}$such that for$\lambda \in ({\lambda }_k,{\lambda }_{k+1})$, problem (1.1) has atleast k pairs of nontrivial solutions.

Note that (f₂) is a weaker version of (g₃). This conditioncombined with (f₁) and the concentration-compactness principle in [14] will allow us to verify that the associated functional satisfies the$(\mathit{PS})$ condition [20] below a fixed level for $\lambda >0$ sufficiently small. Conditions (f₃) and(f₄) provide the geometry required by the symmetric mountain pass theorem [20]. Compared with (g₂), there is no condition imposed on fnear zero in Theorem 1.1. Furthermore, we should mention that our Theorem 1.1 improvesthe main result found in [21]. In that paper, the authors considered only the case where$p(x)$ is constant, while in our present paper, we have showedthat the main result found in [21] is still true for a large class of $p(x)$ functions.

The paper is organized as follows. In Section 2, we introduce some necessarypreliminary knowledge. Section 3 contains the proof of our main result.

2 Preliminaries

We recall some definitions and basic properties of the generalized Lebesgue-Sobolevspaces $L^{p(x)}(\mathrm{\Omega })$ and $W_0^{1,p(x)}(\mathrm{\Omega })$, where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary. And Cwill denote generic positive constants which may vary from line to line.

Set

$$C_{+}(\overline{\mathrm{\Omega }})=\{p(x)\in C(\overline{\mathrm{\Omega }}):p(x)>1,\mathrm{\forall }x\in \overline{\mathrm{\Omega }}\}.$$

For any $p(x)\in C_{+}(\overline{\mathrm{\Omega }})$, we define the variable exponent Lebesgue space

$$L^{p(x)}(\mathrm{\Omega })=\{u\in M(\mathrm{\Omega }):{\int }_{\mathrm{\Omega }}{|u(x)|}^{p(x)}dx<\mathrm{\infty }\}$$

with the norm

$${|u|}_{p(x)}=inf\{\mu >0:{\int }_{\mathrm{\Omega }}{\left|\frac{u}{\mu }\right|}^{p(x)}dx\le 1\},$$

where $M(\mathrm{\Omega })$ is the set of all measurable real functions defined onΩ.

Define the space

$$W^{1,p(x)}(\mathrm{\Omega })=\{u\in L^{p(x)}(\mathrm{\Omega }):|\mathrm{\nabla }u|\in L^{p(x)}(\mathrm{\Omega })\}$$

with the norm

$${\parallel u\parallel }_{1,p(x)}={|u|}_{p(x)}+{|\mathrm{\nabla }u|}_{p(x)}.$$

By $W_0^{1,p(x)}(\mathrm{\Omega })$, we denote the subspace of $W^{1,p(x)}(\mathrm{\Omega })$ which is the closure of $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ with respect to the norm ${\parallel u\parallel }_{1,p(x)}$. Further, we have

Lemma 2.1[22, 23]

There is a constant$C>0$such that for all$u\in W_0^{1,p(x)}(\mathrm{\Omega })$,

$${|u|}_{p(x)}\le C{|\mathrm{\nabla }u|}_{p(x)}.$$

So, ${|\mathrm{\nabla }u|}_{p(x)}$ and ${\parallel u\parallel }_{1,p(x)}$ are equivalent norms in $W_0^{1,p(x)}(\mathrm{\Omega })$. Hence we will use the norm $\parallel u\parallel ={|\mathrm{\nabla }u|}_{p(x)}$ for all $u\in W_0^{1,p(x)}(\mathrm{\Omega })$.

Lemma 2.2[22, 23]

Set$\rho (u)={\int }_{\mathrm{\Omega }}{|u|}^{p(x)}dx$. For$u,u_n\in L^{p(x)}(\mathrm{\Omega })$, we have:

• (1) ${|u|}_{p(x)}<1(=1;>1)\iff \rho (u)<1(=1;>1)$.

• (2) If${|u|}_{p(x)}>1$, then${|u|}_{p(x)}^{p^{-}}\le \rho (u)\le {|u|}_{p(x)}^{p^{+}}$.

• (3) If${|u|}_{p(x)}<1$, then${|u|}_{p(x)}^{p^{+}}\le \rho (u)\le {|u|}_{p(x)}^{p^{-}}$.

• (4) ${lim}_{n\to \mathrm{\infty }}{u}_n=u\iff {lim}_{n\to \mathrm{\infty }}\rho (u_n-u)=0$.

• (5) ${lim}_{n\to \mathrm{\infty }}{|u_n|}_{p(x)}=\mathrm{\infty }\iff {lim}_{n\to \mathrm{\infty }}\rho (u_n)=\mathrm{\infty }$.

Lemma 2.3[23]

If$p_1(x),p_2(x)\in C_{+}(\overline{\mathrm{\Omega }})$with$p_1(x)\le p_2(x)$a.e. in Ω, then thereexists the continuous embedding$L^{p_2(x)}(\mathrm{\Omega })\hookrightarrow L^{p_1(x)}(\mathrm{\Omega })$.

Lemma 2.4[22]

If$q(x)\in C_{+}(\overline{\mathrm{\Omega }})$and$q(x)<p^{\ast }(x)$for any$x\in \mathrm{\Omega }$, the embedding$W^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{q(x)}(\mathrm{\Omega })$is compact.

Lemma 2.5[23]

The conjugate space of$L^{p(x)}(\mathrm{\Omega })$is$L^{p^{\prime }(x)}(\mathrm{\Omega })$, where$\frac{1}{p(x)}+\frac{1}{p^{\prime }(x)}=1$. For any$u\in L^{p(x)}(\mathrm{\Omega })$and$v\in L^{p^{\prime }(x)}(\mathrm{\Omega })$,

$${\int }_{\mathrm{\Omega }}|uv|dx\le (\frac{1}{p^{-}}+\frac{1}{p^{\prime -}}){|u|}_{p(x)}{|v|}_{p^{\prime }(x)}.$$

The energy functional corresponding to problem (1.1) is defined on$W_0^{1,p(x)}(\mathrm{\Omega })$ as follows:

$$I_{\lambda }(u)={\int }_{\mathrm{\Omega }}\frac{1}{p(x)}{|\mathrm{\nabla }u|}^{p(x)}dx-\lambda {\int }_{\mathrm{\Omega }}\frac{1}{q(x)}{|u|}^{q(x)}dx-{\int }_{\mathrm{\Omega }}F(x,u)dx.$$

(2.1)

Then $I_{\lambda }\in C^1(W_0^{1,p(x)}(\mathrm{\Omega }),\mathbb{R})$ and $\mathrm{\forall }u,\varphi \in W_0^{1,p(x)}(\mathrm{\Omega })$,

$$〈I_{\lambda }^{\prime }(u),\varphi 〉={\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{p(x)-2}\mathrm{\nabla }u\mathrm{\nabla }\varphi dx-\lambda {\int }_{\mathrm{\Omega }}{|u|}^{q(x)-2}u\varphi dx-{\int }_{\mathrm{\Omega }}f(x,u)\varphi dx.$$

We say that $u\in W_0^{1,p(x)}(\mathrm{\Omega })$ is a weak solution of problem (1.1) in the weak sense iffor any $\varphi \in W_0^{1,p(x)}(\mathrm{\Omega })$,

$${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{p(x)-2}\mathrm{\nabla }u\mathrm{\nabla }\varphi dx-\lambda {\int }_{\mathrm{\Omega }}{|u|}^{q(x)-2}u\varphi dx-{\int }_{\mathrm{\Omega }}f(x,u)\varphi dx=0.$$

So, the weak solution of problem (1.1) coincides with the critical point of$I_{\lambda }$. Next, we need only to consider the existence of criticalpoints of $I_{\lambda }(u)$.

We say that $I_{\lambda }(u)$ satisfies the ${(\mathit{PS})}_c$ condition if any sequence $\{u_n\}\subseteq W_0^{1,p(x)}(\mathrm{\Omega })$, such that $I_{\lambda }(u_n)\to c$ and $I_{\lambda }^{\prime }(u_n)\to 0$ as $n\to \mathrm{\infty }$, possesses a convergent subsequence. In this article, weshall be using the following version of the symmetric mountain pass theorem [20].

Lemma 2.6[20]

Let$E=V\oplus X$, where E is a real Banach spaceand V is finite dimensional. Supposethat$I\in C^1(E,\mathbb{R})$is an even functionalsatisfying$I(0)=0$and

• (i) there is a constant$\rho >0$such that$I_{\partial B_{\rho }\cap X}\ge 0$;

• (ii) there is a subspace W of E with$dimV<dimW<\mathrm{\infty }$and there is$M>0$such that${max}_{u\in W}I(u)<M$;

• (iii) considering$M>0$given by (ii), I satisfies${(\mathit{PS})}_c$for$0\le c\le M$.

Then I possesses at least$dimW-dimV$pairs of nontrivial critical points.

Next we would use the concentration-compactness principle for variable exponent spaces.This will be the keystone that enables us to verify that $I_{\lambda }$ satisfies the ${(\mathit{PS})}_c$ condition.

Lemma 2.7[14]

Let $q(x)$ and $p(x)$ be two continuous functions such that

$$1<\underset{x\in \overline{\mathrm{\Omega }}}{inf}p(x)\le \underset{x\in \overline{\mathrm{\Omega }}}{sup}p(x)<N\mathit{\text{and}}1\le q(x)\le p^{\ast }(x)\mathit{\text{in }}\mathrm{\Omega }.$$

Let$\{u_n\}$be a weakly convergent sequencein$W_0^{1,p(x)}(\mathrm{\Omega })$with weak limit u such that:

• ${|\mathrm{\nabla }{u}_n|}^{p(x)}\rightharpoonup \mu$weakly in the sense of measures;

• ${|u_n|}^{q(x)}\rightharpoonup \nu$weakly in the sense of measures.

Also assume that$\mathcal{A}=\{x\in \mathrm{\Omega }:q(x)=p^{\ast }(x)\}$is nonempty. Then, for some countableindex set K, we have:

$$\nu ={|u|}^{q(x)}+\sum _{i\in K}{\nu }_i{\delta }_{x_i},{\nu }_i\ge 0,$$

(2.2)

$$\mu \ge {|\mathrm{\nabla }u|}^{p(x)}+\sum _{i\in K}{\mu }_i{\delta }_{x_i},{\mu }_i\ge 0,$$

(2.3)

$$S{\nu }_i^{1/p^{\ast }(x_i)}\le {\mu }_i^{1/p(x_i)}\mathrm{\forall }i\in K,$$

(2.4)

where${\{x_i\}}_{i\in K}\subset \mathcal{A}$and S is the best constant in theGagliardo-Nirenberg-Sobolev inequality for variable exponents,namely

$$S=S_q(\mathrm{\Omega }):=\underset{\varphi \in C_0^{\mathrm{\infty }}(\mathrm{\Omega })}{inf}\frac{\parallel \varphi \parallel }{{|\varphi |}_{q(x)}}.$$

(2.5)

3 Proof of main results

Lemma 3.1 Assume that f satisfies (f₁)and (f₂) with$p^{+}<q^{-}$. Then, given$M>0$, there exists${\lambda }_{\ast }>0$such that$I_{\lambda }$satisfies the${(\mathit{PS})}_c$condition for all$c<M$, provided$0<\lambda <{\lambda }_{\ast }$.

Proof (1) The boundedness of the ${(\mathit{PS})}_c$ sequence.

Let $\{u_n\}$ be a ${(\mathit{PS})}_c$ sequence, i.e., $\{u_n\}$ satisfies $I_{\lambda }(u_n)\to c$, and $I_{\lambda }^{\prime }(u_n)\to 0$ as $n\to \mathrm{\infty }$. If $\parallel u_n\parallel \le 1$, we have done. So we only need to consider the case that$\parallel u_n\parallel >1$ with ${|u_n|}_{q(x)}>1$. We know that

$$\begin{array}{r}{I}_{\lambda }(u_n)={\int }_{\mathrm{\Omega }}\frac{1}{p(x)}{|\mathrm{\nabla }{u}_n|}^{p(x)}dx-\lambda {\int }_{\mathrm{\Omega }}\frac{1}{q(x)}{|u_n|}^{q(x)}dx-{\int }_{\mathrm{\Omega }}F(x,u_n)dx,\\ 〈I_{\lambda }^{\prime }(u_n),u_n〉={\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_n|}^{p(x)}dx-\lambda {\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx-{\int }_{\mathrm{\Omega }}f(x,u_n)u_{n}dx.\end{array}$$

(3.1)

From (f₂), we get

$$\begin{array}{rcl}c+1+\parallel u_n\parallel & \ge & I_{\lambda }(u_n)-\frac{1}{p^{+}}〈I_{\lambda }^{\prime }(u_n),u_n〉\\ \ge & (\frac{1}{p^{+}}-\frac{1}{q^{-}})\lambda {\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx+{\int }_{\mathrm{\Omega }}(\frac{1}{p^{+}}f(x,u_n)u_n-F(x,u_n))dx\\ \ge & (\frac{1}{p^{+}}-\frac{1}{q^{-}})\lambda {\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx-a_1|\mathrm{\Omega }|-a_2{\int }_{\mathrm{\Omega }}{|u_n|}^{\sigma }dx.\end{array}$$

Notice that $q^{-}\le q(x)$, $\mathrm{\forall }x\in \overline{\mathrm{\Omega }}$, then from Lemmas 2.3, 2.4, $W_0^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{q(x)}(\mathrm{\Omega })\hookrightarrow L^{q^{-}}(\mathrm{\Omega })$, so ${|u|}_{q^{-}}\le C_1{|u|}_{q(x)}\le C\parallel u\parallel$. Let $\alpha =(q^{-}-\sigma )/q^{-}$, then $0<\alpha <1$, and from the Hölder inequality,

$$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}{|u_n|}^{\sigma }dx& \le & {\left({\int }_{\mathrm{\Omega }}{|u_n|}^{q^{-}}dx\right)}^{\frac{\sigma }{q^{-}}}{|\mathrm{\Omega }|}^{\frac{q^{-}-\sigma }{q^{-}}}\\ =& {\left({\int }_{\mathrm{\Omega }}{|u_n|}^{q^{-}}dx\right)}^{(1-\alpha )}{|\mathrm{\Omega }|}^{\alpha }\\ \le & {|\mathrm{\Omega }|}^{\alpha }{C}^{(1-\alpha )q^{-}}{\parallel u_n\parallel }^{(1-\alpha )q^{-}}.\end{array}$$

In addition, from Lemma 2.2(2), we can also obtain that

$$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}{|u_n|}^{\sigma }dx& \le & {\left({\int }_{\mathrm{\Omega }}{|u_n|}^{q^{-}}dx\right)}^{(1-\alpha )}{|\mathrm{\Omega }|}^{\alpha }\\ \le & {|\mathrm{\Omega }|}^{\alpha }{\left(C_1{|u_n|}_{q(x)}\right)}^{(1-\alpha )q^{-}}\\ \le & {|\mathrm{\Omega }|}^{\alpha }{C}_1^{(1-\alpha )q^{-}}{\left({\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx\right)}^{(1-\alpha )}.\end{array}$$

Then

$$\begin{array}{r}{I}_{\lambda }(u_n)-\frac{1}{p^{+}}〈I_{\lambda }^{\prime }(u_n),u_n〉\\ \ge (\frac{1}{p^{+}}-\frac{1}{q^{-}})\lambda {\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx-a_1|\mathrm{\Omega }|-a_2{|\mathrm{\Omega }|}^{\alpha }{C}_1^{(1-\alpha )q^{-}}{\left({\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx\right)}^{(1-\alpha )},\end{array}$$

(3.2)

and

$$\begin{array}{rcl}c+1+\parallel u_n\parallel & \ge & I_{\lambda }(u_n)-\frac{1}{p^{+}}〈I_{\lambda }^{\prime }(u_n),u_n〉\\ \ge & (\frac{1}{p^{+}}-\frac{1}{q^{-}})\lambda {\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx-a_1|\mathrm{\Omega }|-C{\parallel u_n\parallel }^{(1-\alpha )q^{-}}.\end{array}$$

So we have

$${\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx\le C+C\parallel u_n\parallel +C{\parallel u_n\parallel }^{(1-\alpha )q^{-}}.$$

(3.3)

From (3.1), (3.3) and (f₁), we have

$$\begin{array}{rcl}\frac{1}{p^{+}}{\parallel u_n\parallel }^{p^{-}}& \le & {\int }_{\mathrm{\Omega }}\frac{1}{p(x)}{|\mathrm{\nabla }{u}_n|}^{p(x)}dx=I_{\lambda }(u_n)+\lambda {\int }_{\mathrm{\Omega }}\frac{1}{q(x)}{|u_n|}^{q(x)}dx+{\int }_{\mathrm{\Omega }}F(x,u_n)dx\\ \le & C+C{\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}dx\\ \le & C+C\parallel u_n\parallel +C{\parallel u_n\parallel }^{(1-\alpha )q^{-}}.\end{array}$$

Noting that $(1-\alpha )q^{-}=\sigma <p^{-}$, we have that $\{u_n\}$ is bounded.

• (2) Up to a subsequence, $u_n\to u$ in $W_0^{1,p(x)}(\mathrm{\Omega })$.

By Lemma 2.7, we can assume that there exist two measures μ,ν and a function $u\in W_0^{1,p(x)}(\mathrm{\Omega })$ such that

$$\begin{array}{c}{u}_n\rightharpoonup u\text{weakly in }{W}_0^{1,p(x)}(\mathrm{\Omega }),\hfill \\ {|\mathrm{\nabla }{u}_n|}^{p(x)}\rightharpoonup \mu \text{weakly in the sense of measures},\hfill \\ {|u_n|}^{q(x)}\rightharpoonup \nu \text{weakly in the sense of measures},\hfill \\ \nu ={|u|}^{q(x)}+\sum _{j\in K}{\nu }_j{\delta }_{x_j},\hfill \\ \mu \ge {|\mathrm{\nabla }u|}^{p(x)}+\sum _{j\in K}{\mu }_j{\delta }_{x_j},\hfill \\ S{\nu }_j^{1/p^{\ast }(x_j)}\le {\mu }_j^{1/p(x_j)}.\hfill \end{array}$$

Choose a function $\phi (x)\in C_0^{\mathrm{\infty }}({\mathbb{R}}^N)$ such that $0\le \phi (x)\le 1$, $\phi (x)\equiv 1$ on $B(0,1)$ and $\phi (x)\equiv 0$ on ${\mathbb{R}}^N\setminus B(0,2)$. For any $x\in {\mathbb{R}}^N$, $\epsilon >0$ and $j\in K$, let ${\phi }_{j,\epsilon }(x)=\phi (\frac{x-x_j}{\epsilon })$. It is clear that $\{{\phi }_{j,\epsilon }{u}_n\}$ is bounded in $W_0^{1,p(x)}(\mathrm{\Omega })$. From $I_{\lambda }^{\prime }(u_n)\to 0$, we can obtain $〈I_{\lambda }^{\prime }(u_n),{\phi }_{j,\epsilon }{u}_n〉\to 0$, as $n\to \mathrm{\infty }$, i.e.,

$$\begin{array}{r}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_n|}^{p(x)-2}\mathrm{\nabla }{u}_n\cdot u_n\mathrm{\nabla }{\phi }_{j,\epsilon }dx+{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_n|}^{p(x)}{\phi }_{j,\epsilon }dx\\ -\lambda {\int }_{\mathrm{\Omega }}{|u_n|}^{q(x)}{\phi }_{j,\epsilon }dx-{\int }_{\mathrm{\Omega }}f(x,u_n)u_n{\phi }_{j,\epsilon }dx\to 0.\end{array}$$

(3.4)

From (f₁), by Lemma 2.7, we have

$$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_n|}^{p(x)-2}\mathrm{\nabla }{u}_n\cdot u_n\mathrm{\nabla }{\phi }_{j,\epsilon }dx\\ =\lambda {\int }_{\mathrm{\Omega }}{\phi }_{j,\epsilon }d\nu -{\int }_{\mathrm{\Omega }}{\phi }_{j,\epsilon }d\mu +{\int }_{\mathrm{\Omega }}f(x,u)u{\phi }_{j,\epsilon }dx.\end{array}$$

(3.5)

By the Hölder inequality, it is easy to check that

$$\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_n|}^{p(x)-2}\mathrm{\nabla }{u}_n\cdot u_n\mathrm{\nabla }{\phi }_{j,\epsilon }dx=0.$$

From (3.5), as $\epsilon \to 0$, we obtain $\lambda {\nu }_j={\mu }_j$. From Lemma 2.7, we conclude that

$${\nu }_j=0\text{or}{\nu }_j\ge S^{N}max\{{\lambda }^{-\frac{N}{p^{+}}},{\lambda }^{-\frac{N}{p^{-}}}\}.$$

(3.6)

Given $M>0$, set

$$\begin{array}{rcl}{\lambda }_{\ast }& =& min\{S^{p^{+}},S^{p^{-}},{\left(\frac{S^N{(\frac{1}{p^{+}}-\frac{1}{q^{-}})}^{\frac{1}{\alpha }}}{{(M+a_1|\mathrm{\Omega }|+a_2{|\mathrm{\Omega }|}^{\alpha }{C}_1^{(1-\alpha )q^{-}})}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\frac{N}{p^{+}}-\frac{1}{\alpha }}},\\ {\left(\frac{S^N{(\frac{1}{p^{+}}-\frac{1}{q^{-}})}^{\frac{1}{\alpha }}}{{(M+a_1|\mathrm{\Omega }|+a_2{|\mathrm{\Omega }|}^{\alpha }{C}_1^{(1-\alpha )q^{-}})}^{\frac{1}{\alpha }}}\right)}^{\frac{1}{\frac{N}{p^{\phantom{a}-}}-\frac{1}{\alpha }}}\},\end{array}$$

where S is given by (2.5). Considering $0<\lambda <{\lambda }_{\ast }$, we have

$$1<S^N{\lambda }^{-\frac{N}{p^{+}}},1<S^N{\lambda }^{-\frac{N}{p^{-}}},$$

(3.7)

and

$${\left(\frac{M+a_1|\mathrm{\Omega }|+a_2{|\mathrm{\Omega }|}^{\alpha }{C}_1^{(1-\alpha )q^{-}}}{(\frac{1}{p^{+}}-\frac{1}{q^{-}})\lambda }\right)}^{\frac{1}{\alpha }}<S^{N}min\{{\lambda }^{-\frac{N}{p^{+}}},{\lambda }^{-\frac{N}{p^{+}}}\}.$$

(3.8)

We claim that ${\int }_{\mathrm{\Omega }}d\nu <S^{N}min\{{\lambda }^{-\frac{N}{p^{+}}},{\lambda }^{-\frac{N}{p^{-}}}\}$. Indeed, if ${\int }_{\mathrm{\Omega }}d\nu \le 1$, this follows by (3.7). Otherwise, taking$n\to \mathrm{\infty }$ in (3.2), we obtain

$$\begin{array}{rcl}(\frac{1}{p^{+}}-\frac{1}{q^{-}})\lambda {\int }_{\mathrm{\Omega }}d\nu & \le & a_1|\mathrm{\Omega }|+a_2{|\mathrm{\Omega }|}^{\alpha }{C}_1^{(1-\alpha )q^{-}}{\left({\int }_{\mathrm{\Omega }}d\nu \right)}^{1-\alpha }+c\\ \le & (M+a_1|\mathrm{\Omega }|+a_2{|\mathrm{\Omega }|}^{\alpha }{C}_1^{(1-\alpha )q^{-}}){\left({\int }_{\mathrm{\Omega }}d\nu \right)}^{1-\alpha }.\end{array}$$

Therefore, by (3.8), the claim is proved. As a consequence of this fact, we concludethat ${\nu }_j=0$ for all $j\in K$. Therefore, $u_n\to u$ in $L^{q(x)}(\mathrm{\Omega })$. Then, with the similar step in [17], we can get that $u_n\to u$ in $W_0^{1,p(x)}(\mathrm{\Omega })$. □

Next we prove Theorem 1.1 by verifying that the functional $I_{\lambda }$ satisfies the hypotheses of Lemma 2.6. First, we recallthat each basis ${\{e_i\}}_{i\in \mathbb{N}}$ for a real Banach space E is a Schauder basis forE, i.e., given $n\in \mathbb{N}$, the functional $e_n^{\ast }:E\to \mathbb{R}$ defined by

$$e_n^{\ast }(v)={\alpha }_n,v=\sum _{i=1}^{\mathrm{\infty }}{\alpha }_i{e}_i\in E$$

is a bounded linear functional [24, 25]. Now, fixing a Schauder basis ${\{e_i\}}_{i\in \mathbb{N}}$ for $W_0^{1,p(x)}(\mathrm{\Omega })$, for $j\in \mathbb{N}$, we set

$$\begin{array}{r}{V}_j=\{u\in W_0^{1,p(x)}(\mathrm{\Omega }):e_i^{\ast }(u)=0,i>j\},\\ X_j=\{u\in W_0^{1,p(x)}(\mathrm{\Omega }):e_i^{\ast }(u)=0,i\le j\},\end{array}$$

(3.9)

then $W_0^{1,p(x)}(\mathrm{\Omega })=V_j\oplus X_j$.

Lemma 3.2 Given$1\le r(x)<p^{\ast }(x)$for all$x\in \mathrm{\Omega }$and$\delta >0$, there is$j\in \mathbb{N}$such that for all$u\in X_j$, ${|u|}_{r(x)}\le \delta \parallel u\parallel$.

Proof We prove the lemma by contradiction. Suppose that there exist$\delta >0$ and $u_j\in X_j$ for every $j\in \mathbb{N}$ such that ${|u_j|}_{r(x)}\ge \delta \parallel u_j\parallel$. Taking $v_j=\frac{u_j}{{|u_j|}_{r(x)}}$, we have ${|v_j|}_{r(x)}=1$ for every $j\in \mathbb{N}$ and $\parallel v_j\parallel \le \frac{1}{\delta }$. Hence $\{v_j\}\subset W_0^{1,p(x)}(\mathrm{\Omega })$ is a bounded sequence, and we may suppose, without loss ofgenerality, that $v_j\rightharpoonup v$ in $W_0^{1,p(x)}(\mathrm{\Omega })$. Furthermore, $e_n^{\ast }(v)=0$ for every $n\in \mathbb{N}$ since $e_n^{\ast }(v_j)=0$ for all $j\ge n$. This shows that $v=0$. On the other hand, by the compactness of the embedding$W_0^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{r(x)}(\mathrm{\Omega })$, we conclude that ${|v|}_{r(x)}=1$. This proves the lemma. □

Lemma 3.3 Suppose that f satisfies (f₃),then there exist$j\in \mathbb{N}$and$\rho ,\alpha ,\tilde{\lambda }>0$such that$I{|}_{\partial B_{\rho }\cap X_j}\ge \alpha$for all$0<\lambda <\tilde{\lambda }$.

Proof Now suppose that $\parallel u\parallel >1$, with ${|u|}_{r(x)}>1$, ${|u|}_{q(x)}>1$. From (f₃), we know that

$$\begin{array}{rcl}{I}_{\lambda }(u)& =& {\int }_{\mathrm{\Omega }}\frac{{|\mathrm{\nabla }u|}^{p(x)}}{p(x)}dx-\lambda {\int }_{\mathrm{\Omega }}\frac{{|u|}^{q(x)}}{q(x)}dx-{\int }_{\mathrm{\Omega }}F(x,u)dx\\ \ge & \frac{1}{p^{+}}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{p(x)}dx-\frac{\lambda }{q^{-}}{\int }_{\mathrm{\Omega }}{|u|}^{q(x)}dx-b_1{\int }_{\mathrm{\Omega }}{|u|}^{r(x)}dx-b_2|\mathrm{\Omega }|.\end{array}$$

Consequently, considering $\delta >0$ to be chosen posteriorly by Lemma 3.2, we have, for all$u\in X_j$ and j sufficiently large,

$$\begin{array}{rcl}{I}_{\lambda }(u)& \ge & \frac{1}{p^{+}}{\parallel u\parallel }^{p^{-}}-\frac{\lambda C}{q^{-}}{\parallel u\parallel }^{q^{+}}-b_1{\delta }^{r^{+}}{\parallel u\parallel }^{r^{+}}-b_2|\mathrm{\Omega }|\\ \ge & {\parallel u\parallel }^{p^{-}}(\frac{1}{p^{+}}-b_1{\delta }^{r^{+}}{\parallel u\parallel }^{r^{+}-p^{-}})-b_2|\mathrm{\Omega }|-\frac{C\lambda }{q^{-}}{\parallel u\parallel }^{q^{+}}.\end{array}$$

Now taking $1<\parallel u\parallel =\rho (\delta )$ such that $b_1{\delta }^{r^{+}}{\rho }^{r^{+}-p^{-}}=\frac{1}{2p^{+}}$ and noting that $r^{+}>p^{-}$, so $\rho (\delta )\to +\mathrm{\infty }$, if $\delta \to 0$. We can choose $\delta >0$ such that $\frac{{\rho }^{p^{-}}}{2p^{+}}-b_1|\mathrm{\Omega }|>\frac{{\rho }^{p^{-}}}{4p^{+}}$. Next, we take $\tilde{\lambda }>0$ such that for $0<\lambda <\tilde{\lambda }$,

$$I_{\lambda }(u)\ge \frac{{\rho }^{p^{-}}}{4p^{+}}-\frac{C\lambda }{q^{-}}{\rho }^{q^{+}}>0$$

for every $u\in X_j$, $\parallel u\parallel =\rho$, the proof is complete. □

Lemma 3.4 Suppose that f satisfies (f₄),then, given$m\in \mathbb{N}$, there exist asubspace W of$W_0^{1,p(x)}(\mathrm{\Omega })$and a constant$M_m>0$such that$dimW=m$and${max}_{u\in W}I(u)<M_m$.

Proof Let $x_0\in {\mathrm{\Omega }}_0$ and $r_0>0$ be such that $\overline{B(x_0,r_0)}\subset \mathrm{\Omega }$, and $0<|\overline{B(x_0,r_0)}\cap {\mathrm{\Omega }}_0|<\frac{|{\mathrm{\Omega }}_0|}{2}$. First, we take $v_1\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ with $supp(v_1)=\overline{B(x_0,r_0)}$. Considering ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_0\setminus [\overline{B(x_0,r_0)}\cap {\mathrm{\Omega }}_0]\subset {\hat{\mathrm{\Omega }}}_0=\overline{\mathrm{\Omega }\setminus B(x_0,r_0)}$, we have $|{\mathrm{\Omega }}_1|\ge \frac{|{\mathrm{\Omega }}_0|}{2}>0$. Let $x_1\in {\mathrm{\Omega }}_1$ and $r_1>0$ such that $\overline{B(x_1,r_1)}\subset {\hat{\mathrm{\Omega }}}_0$, and $0<|\overline{B(x_1,r_1)}\cap {\mathrm{\Omega }}_1|<\frac{|{\mathrm{\Omega }}_1|}{2}$. Next, we take $v_2\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ with $supp(v_2)=\overline{B(x_1,r_1)}$. After a finite number of steps, we get$v_1,v_2,\dots ,v_m$ such that $supp(v_i)\cap supp(v_j)=\mathrm{\varnothing }$, $i\ne j$, and $|supp(v_j)\cap {\mathrm{\Omega }}_0|>0$ for all $i,j\in \{1,2,\dots ,m\}$. Let $W=span\{v_1,v_2,\dots ,v_m\}$, by construction, $dimW=m$, and for every $v\in W\setminus \{0\}$,

$${\int }_{{\mathrm{\Omega }}_0}{|v|}^{p^{+}}dx>0.$$

Since

$$\underset{u\in W\setminus \{0\}}{max}{I}_0(u)=\underset{t>0,v\in W\cap \partial B_1(0)}{max}({\int }_{\mathrm{\Omega }}\frac{{(t|\mathrm{\nabla }v|)}^{p(x)}}{p(x)}dx-{\int }_{\mathrm{\Omega }}F(x,tv)dx),$$

consider the case that $t>1$, then $I_0(tv)\le \frac{t^{p^{+}}}{p^{-}}-{\int }_{\mathrm{\Omega }}F(x,tv)dx=t^{p^{+}}(\frac{1}{p^{-}}-\frac{1}{t^{p^{+}}}{\int }_{\mathrm{\Omega }}F(x,tv)dx)$. Now it suffices to verify that

$$\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t^{p^{+}}}{\int }_{\mathrm{\Omega }}F(x,tv)dx>\frac{1}{p^{-}}.$$

From condition (f₄), given $L>0$, there is $C>0$ such that for every $s\in \mathbb{R}$, a.e. x in ${\mathrm{\Omega }}_0$,

$$F(x,s)\ge L{|s|}^{p^{+}}-C.$$

Consequently, for $v\in \partial B_1(0)\cap W$ and $t>1$,

$${\int }_{\mathrm{\Omega }}F(x,tv)dx\ge L{t}^{p^{+}}{\int }_{{\mathrm{\Omega }}_0}{|v|}^{p^{+}}dx-C{t}^{p^{+}}{\int }_{\mathrm{\Omega }\setminus {\mathrm{\Omega }}_0}{h}_1(x){|v|}^{p(x)}dx-C_2$$

and

$$\underset{t\to \mathrm{\infty }}{lim}\frac{{\int }_{\mathrm{\Omega }}F(x,tv)dx}{t^{p^{+}}}\ge L{\int }_{{\mathrm{\Omega }}_0}{|v|}^{p^{+}}dx-C{\int }_{\mathrm{\Omega }\setminus {\mathrm{\Omega }}_0}{h}_1(x){|v|}^{p(x)}dx\ge Lr-CR,$$

where $r=min\{{\int }_{{\mathrm{\Omega }}_0}{|v|}^{p^{+}}dx,v\in \partial B_1(0)\cap W\}$ and $R=max\{{\int }_{\mathrm{\Omega }\setminus {\mathrm{\Omega }}_0}{h}_1(x){|v|}^{p(x)}dx,v\in \partial B_1(0)\cap W\}$. Observing that W is finite dimensional, we have$R<+\mathrm{\infty }$, $r>0$, and the inequality is obtained by taking$L>\frac{1}{r}(\frac{1}{p^{-}}+CR)$. The proof is complete. □

Proof of Theorem 1.1 First, we recall that $W_0^{1,p(x)}(\mathrm{\Omega })=V_j\oplus X_j$, where $V_j$ and $X_j$ are defined in (3.9). Invoking Lemma 3.3, we find$j\in \mathbb{N}$, and $I_{\lambda }$ satisfies (i) with $X=X_j$. Now, by Lemma 3.4, there is a subspace W of$W_0^{1,p(x)}(\mathrm{\Omega })$ with $dimW=k+j=k+dim{V}_j$ such that $I_{\lambda }$ satisfies (ii). By Lemma 3.1, $I_{\lambda }$ satisfies (iii). Since $I_{\lambda }(0)=0$ and $I_{\lambda }$ is even, we may apply Lemma 2.6 to conclude that$I_{\lambda }$ possesses at least k pairs of nontrivial criticalpoints. The proof is complete. □