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Nonlinear boundary value problems of a class of elliptic equations involving critical variable exponents

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Abstract

In this paper, we first obtain the existence of solutions for a class of elliptic equations involving critical variable exponents and nonlinear boundary values by the mountain pass theorem and concentration compactness principle. Then, under suitable assumptions, we obtain a sequence of solutions with positive energies going towards infinity by Fountain Theorem.

Introduction

In the studies of electrorheological fluids, nonlinear elasticity, and image restoration in practical applications, the classical Lebesgue and Sobolev spaces are inapplicable; see [1,2,3]. Such problems are inhomogeneous and nonlinear with variable exponential growth conditions. So we need to study the problems based on the theory of variable exponent Lebesgue and Sobolev spaces.

Since Kováčik and Rákosník first studied the \(L^{p(x)}\) spaces and \(W^{k,p(x)}\) spaces in [4], a lot of research has been done concerning these kinds of variable exponent spaces. The existence of solutions for \(p(x)\)-Laplacian Dirichlet problems on bounded domains has been widely discussed. For example, in [5] and [6], some results as regards the existence of solutions under some conditions are obtained.

The nonlinear elliptic boundary value problems appear when we study the conformal deformations on Riemannian manifolds with boundary. The study of nonlinear elliptic boundary value problems with p-Laplacian has become an interesting topic in recent years. Many results have been obtained on this kind of problems; see [7,8,9]. In the fractional Laplacian setting, the existence of solutions for the problem has been obtained; see [10,11,12,13,14].

But at present there are few papers on the study of nonlinear elliptic boundary value problems with \(p(x)\)-Laplacian. So this topic is worth further discussing.

In this paper, we consider the problem

$$\begin{aligned} \textstyle\begin{cases} \operatorname{div}(a(x),\nabla {u})+ \vert u \vert ^{p(x)-2}u=f(x,u)+h(x) \vert u \vert ^{p^{*}(x)-2}u, \quad x\in \varOmega , \\ a(x,\nabla {u})\cdot {v(x)}=b(x,u), \quad x\in \partial \varOmega , \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(\varOmega \subset \mathbb{R}^{N}\) is a bounded domain with smooth boundary, \(p(x)\) is Lipschitz continuous and satisfies \(1< p_{1} \leq p(x)\leq p_{2}<N\), \(p^{*}(x)=\frac{Np(x)}{N-p(x)}\). We assume that \(a:\overline{\varOmega }\times \mathbb{R}^{N}\rightarrow \mathbb{R}\) is a Carathéodory function and we have the continuous derivative with respect to η of a function \(A:\overline{\varOmega }\times \mathbb{R}^{N}\rightarrow \mathbb{R}\). Suppose that a and A satisfy the following hypotheses:

(A1):

For \(\forall x\in \overline{\varOmega }\), the equality \(A(x,0)=0\) holds.

(A2):

There exists a positive constant \(c_{0}\) such that

$$ \bigl\vert a(x,\eta ) \bigr\vert \leq c_{0} \bigl(1+ \vert \eta \vert ^{p(x)-1} \bigr) $$

for all \(x\in \overline{\varOmega }\) and \(\eta \in \mathbb{R}^{N}\).

(A3):

For all \(x\in \overline{\varOmega }\) and \(\eta _{1},\eta _{2}\in \mathbb{R}^{N}\), the following inequality holds:

$$ 0\leq \bigl[a(x,\eta _{1})-a(x,\eta _{2}) \bigr]\cdot ( \eta _{1}-\eta _{2}), $$

where equality holds if and only if \(\eta _{1}=\eta _{2}\).

(A4):

For all \(x\in \overline{\varOmega }\) and \(\eta \in \mathbb{R}^{N}\), the inequalities

$$ \vert \eta \vert ^{p(x)}\leq a(x,\eta )\cdot \eta \leq p(x)A(x,\eta ) $$

hold true.

(A5):

For all \(x\in \overline{\varOmega }\) and \(\eta \in \mathbb{R}^{N}\), the equality \(A(x,-\eta )=A(x,\eta )\) holds true.

The above type of assumptions can be found in other papers too; for example, see [15, 16]. But in [16], the authors establish the existence of a solution for an elliptic problem with Dirichlet boundary conditions, and in [15], the authors consider the subcritical case. In the present paper, the problem involves not only the critical Sobolev exponents, but also the nonlinear boundary conditions. Because of the critical exponents, the compactness of the embedding fails, so to recover the loss of the compactness, we use the concentration compactness principle in [17].

Throughout this paper, we assume that the following conditions hold:

(F1):

\(f\in C(\overline{\varOmega }\times \mathbb{R})\), \(f(x,0)\equiv 0\) and \(\vert f(x,t)\vert \leq C_{1}(1+\vert t\vert ^{\alpha _{1}(x)-1})\), \(\alpha _{1}\in C(\overline{ \varOmega })\) with \(p(x)\ll \alpha _{1}(x)\ll p^{*}(x)\), and \(F(x,t)>0\) in \(\varOmega _{0}\times \mathbb{R}\) for some nonempty open set \(\varOmega _{0} \subset \varOmega \), where \(C_{1}\) is a positive constant.

(F̃1):

\(f\in C(\overline{\varOmega }\times \mathbb{R})\), \(\vert f(x,t)\vert \leq C_{1}(1+\vert t\vert ^{\alpha _{1}(x)-1})\), \(\alpha _{1}\in C(\overline{\varOmega })\) with \(1\leq \alpha _{1}(x)\ll p(x)\) and \(F(x,t)>0\) in \(\varOmega _{0}\times \mathbb{R}\) for some nonempty open set \(\varOmega _{0}\subset \varOmega \).

(F2):

\(f(x,t)=-f(x,-t)\) for any \((x,t)\in \overline{\varOmega }\times \mathbb{R}\).

(F3):

For any \((x,t)\in \overline{\varOmega }\times \mathbb{R}\), there exists a function \(\mu _{1}(x)\in C^{1}(x)\) such that \(\mu _{1}(x) \gg p(x)\) and \(0\leq \mu _{1}(x)F(x,t)\leq f(x,t)t\), where \(F(x,t)=\int ^{t}_{0}f(x,s)\,ds\).

(F4):

\(f(x,t)=o(\vert t\vert ^{p(x)-1})\) hold uniformly for any \(x\in \overline{ \varOmega }\), as \(t\rightarrow 0\).

(B1):

\(b\in C(\overline{\varOmega }\times \mathbb{R})\), \(b(x,0)\equiv 0\) and \(\vert b(x,t)\vert \leq C_{2}\vert t\vert ^{\alpha _{2}(x)-1}\), \(\alpha _{2}\in C(\overline{ \varOmega })\) with \(p(x)\ll \alpha _{2}(x)\ll p_{*}(x)\), and \(B(x,t)>0\) in \(\partial \varOmega \times \mathbb{R}\), where \(C_{2}\) is a positive constant and \(p_{*}(x)=\frac{(N-1)p(x)}{N-p(x)}\).

(B̃1):

\(b\in C(\overline{\varOmega }\times \mathbb{R})\), \(\vert b(x,t)\vert \leq C_{2}(1+\vert t\vert ^{\alpha _{2}(x)-1})\), \(\alpha _{2}\in C(\overline{\varOmega })\) with \(p(x)\ll \alpha _{2}(x) \ll p_{*}(x)\) and \(B(x,t)>0\) in \(\partial \varOmega \times \mathbb{R}\).

(B2):

\(b(x,t)=-b(x,-t)\) for any \((x,t)\in \overline{\varOmega }\times \mathbb{R}\).

(B3):

For any \((x,t)\in \partial \varOmega \times \mathbb{R}\), there exists a function \(\mu _{2}(x)\in C^{1}(x)\) such that \(\mu _{2}(x)\gg p(x)\) and \(0\leq \mu _{2}(x)B(x,t)\leq b(x,t)t\), where \(B(x,t)=\int ^{t}_{0}b(x,s)\,ds\).

(H1):

For any \(x\in \varOmega \), there exists \(h_{1}>0\) such that \(h(x)\geq h_{1}\) and \(h(x)\in L^{\infty }(\varOmega )\).

Preliminaries

We first recall some facts on spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). For details see [4, 18, 19].

Let \(\mathbf{P}(\varOmega )\) be the set of all Lebesgue measurable functions \(p:\varOmega \to [1,\infty ]\), we denote

$$ \rho _{p(x)}(u)= \int _{\varOmega \setminus \varOmega _{\infty }} \vert u \vert ^{p(x)} \,dx+ \sup _{x\in \varOmega _{\infty }} \bigl\vert u(x) \bigr\vert , $$

where \(\varOmega _{\infty }=\{x\in \varOmega :p(x)=\infty \}\).

The variable exponent Lebesgue space \(L^{p(x)}(\varOmega )\) is the class of all functions u such that \(\rho _{p(x)}(tu)<\infty \), for some \(t>0\). \(L^{p(x)}(\varOmega )\) is a Banach space equipped with the norm

$$ \Vert u \Vert _{L^{p(x)}}=\inf \biggl\{ \lambda >0:\rho _{p(x)} \biggl( \frac{u}{\lambda } \biggr) \leq 1 \biggr\} . $$

For any \(p\in \mathbf{P}(\varOmega )\), we define the conjugate function \(p'(x)\) as

$$ p'(x)= \textstyle\begin{cases} \infty , & x\in \varOmega _{1}=\{x\in \varOmega :p(x)=1\}, \\ 1, & x\in \varOmega _{\infty }, \\ \frac{p(x)}{p(x)-1}, & x\in \varOmega \setminus (\varOmega _{1}\cup \varOmega _{\infty }). \end{cases} $$

Theorem 2.1

Let \(p\in \mathbf{P}(\varOmega )\). For any \(u\in L^{p(x)}(\varOmega )\) and \(v\in L^{p'(x)}(\varOmega )\),

$$ \int _{\varOmega } \vert uv \vert \,dx\leq 2 \Vert u \Vert _{L^{p(x)}} \Vert v \Vert _{L^{p'(x)}}. $$

For any \(p\in \mathbf{P}(\varOmega )\), we denote

$$ p_{1}=\inf_{x\in \varOmega } p(x),\quad \quad p_{2}=\sup_{x \in \varOmega } p(x), $$

and we denote by \(p(x)\ll q(x)\) the fact that \(\inf_{x\in \varOmega } (q(x)-p(x))>0\).

Theorem 2.2

Let \(p\in \mathbf{P}(\varOmega )\) with \(p_{2}<\infty \). For any \(u\in L^{p(x)}(\varOmega )\), we have

  1. (1)

    if \(\Vert u\Vert _{L^{p(x)}}\geq 1\), then \(\Vert u\Vert _{L^{p(x)}}^{p_{1}} \leq \int _{\varOmega } \vert u\vert ^{p(x)} \,dx\leq \Vert u\Vert _{L^{p(x)}}^{p_{2}}\);

  2. (2)

    if \(\Vert u\Vert _{L^{p(x)}}<1\), then \(\Vert u\Vert _{L^{p(x)}}^{p_{2}} \leq \int _{\varOmega } \vert u\vert ^{p(x)} \,dx\leq \Vert u\Vert _{L^{p(x)}}^{p_{1}}\).

The variable exponent Sobolev space \(W^{1,p(x)}(\varOmega )\) is the class of all functions \(u\in L^{p(x)}(\varOmega )\) such that \(\vert \nabla u\vert \in L ^{p(x)}(\varOmega )\). \(W^{1,p(x)}(\varOmega )\) is a Banach space equipped with the norm

$$ \Vert u \Vert _{W^{1,p(x)}}= \Vert u \Vert _{L^{p(x)}}+ \Vert \nabla u \Vert _{L^{p(x)}}. $$

For \(u\in W^{1,p(x)}(\varOmega )\), if we define

$$ |\!|\!|u |\!|\!|=\inf \biggl\{ t>0: \int _{\varOmega } \frac{ \vert u \vert ^{p(x)}+ \vert \nabla u \vert ^{p(x)}}{t ^{p(x)}} \,dx\leq 1 \biggr\} , $$

then \(|\!|\!|\cdot |\!|\!|\) and \(\Vert \cdot \Vert _{W^{1,p(x)}}\) are equivalent norms on \(W^{1,p(x)}(\varOmega )\). In fact, we have

$$ \frac{1}{2} \Vert u \Vert _{W^{1,p(x)}}\leq |\!|\!|u |\!|\!|\leq 2 \Vert u \Vert _{W^{1,p(x)}}. $$

Theorem 2.3

For any \(u\in W^{1,p(x)}(\varOmega )\), we have

  1. (1)

    if \(|\!|\!|u|\!|\!|\geq 1\), then \(|\!|\!|u|\!|\!|^{p_{1}}\leq \int _{\varOmega } (\vert \nabla u\vert ^{p(x)}+\vert u\vert ^{p(x)}) \,dx\leq |\!|\!|u|\!|\!|^{p_{2}}\);

  2. (2)

    if \(|\!|\!|u|\!|\!|<1\), then \(|\!|\!|u|\!|\!|^{p_{2}}\leq \int _{\varOmega } (\vert \nabla u\vert ^{p(x)}+\vert u\vert ^{p(x)}) \,dx\leq |\!|\!|u|\!|\!|^{p_{1}}\).

Theorem 2.4

Let Ω be a bounded domain with the cone property. If \(p\in C(\bar{\varOmega })\) satisfying \(1< p_{1}\leq p(x)\leq p_{2}<N\) and q is a measurable function defined on Ω with

$$ p(x)\leq q(x)\ll p^{*}(x),\quad \textit{a.e. } x\in \varOmega , $$

then the embedding \(W^{1,p(x)}(\varOmega )\hookrightarrow L^{q(x)}( \varOmega )\) is compact.

Theorem 2.5

Let Ω be a domain with the cone property. If p is Lipschitz continuous and satisfies \(1< p_{1}\leq p(x)\leq p_{2}<N\), q is a measurable function defined on Ω with

$$ p(x)\leq q(x)\leq p^{*}(x),\quad \textit{a.e. }x\in \varOmega , $$

then the embedding \(W^{1,p(x)}(\varOmega )\hookrightarrow L^{q(x)}( \varOmega )\) is continuous.

Theorem 2.6

Let \(\varOmega \subset \mathbb{R}^{N}\) be an open bounded domain with Lipschitz boundary. Suppose that \(p\in C(\bar{\varOmega })\) and \(1< p_{1}\leq p(x)\leq p_{2}<N\). If \(q\in C(\partial \varOmega )\) satisfies the condition

$$ 1\leq q(x)< p_{*}(x),\quad \forall x\in \partial \varOmega , $$

then the boundary trace embedding \(W^{1,p(x)}(\varOmega )\rightarrow L ^{q(x)}(\partial \varOmega )\) is compact.

In the proof of the main results, we will use the following principle of concentration compactness in \(W^{1,p(x)}(\varOmega )\), established in [17].

Theorem 2.7

Assume that p is Lipschitz continuous on Ω̄ and satisfies \(1< p_{1}\leq p(x)\leq p_{2}<N\), and Ω is a bounded domain in \(\mathbb{R}^{N}\). Let \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) with \(\Vert \nabla u_{n}\Vert _{L^{p(x)}}\leq 1\) such that

$$\begin{aligned}& u_{n}\to u \quad \textit{weakly in }W^{1,p(x)}(\varOmega ), \\& \vert \nabla u_{n} \vert ^{p(x)}\to \mu \quad \textit{weak-}*\textit{ in } M(\bar{\varOmega }), \\& \vert u_{n} \vert ^{p^{*}(x)}\to \nu \quad \textit{weak-}* \textit{ in } M( \bar{\varOmega }), \end{aligned}$$

as \(n\to \infty \). Denote

$$ C_{*}=\sup \biggl\{ \int _{\varOmega } \vert u \vert ^{p^{*}(x)} \,dx: \Vert \nabla u_{n} \Vert _{L ^{p(x)}}\leq 1, u\in W^{1,p(x)}( \varOmega ) \biggr\} . $$

Then the limit measures are of the form

$$\begin{aligned}& \mu = \vert \nabla u \vert ^{p(x)}+\sum_{j\in J} \mu _{j} \delta _{x_{j}} + \widetilde{\mu },\quad\quad \mu ( \bar{\varOmega })\leq 1, \\& \nu = \vert u \vert ^{p^{*}(x)}+\sum_{j\in J} \nu _{j} \delta _{x_{j}}, \quad\quad \nu (\bar{\varOmega }) \leq C_{*}, \end{aligned}$$

where J is a countable set, \(\{\mu _{j}\},\{\nu _{j}\}\subset [0, \infty )\), \(\{x_{j}\}\subset \bar{\varOmega }\), \(\widetilde{\mu }\in M( \varOmega )\) is a non-atomic nonnegative measure. The atoms and the regular part satisfy the generalized Sobolev inequality

$$ \begin{gathered} \nu (\bar{\varOmega }) \leq C_{*}\max \bigl\{ \mu (\bar{\varOmega }) ^{p^{*}_{2}/p _{1}},\mu (\bar{ \varOmega })^{p^{*}_{1}/p_{2}} \bigr\} , \\ \nu _{j}\leq C_{*}\max \bigl\{ \mu _{j}^{p^{*}_{2}/p_{1}}, \mu _{j}^{p^{*}_{1}/p _{2}} \bigr\} , \quad \forall j\in J, \end{gathered} $$
(2.1)

where \(p^{*}_{1}=\inf_{x\in \varOmega } p^{*}(x)\), \(p^{*}_{2}=\sup_{x \in \varOmega } p^{*}(x)\).

Existence of solutions for the problems

Set

$$\begin{aligned}& \varLambda (u)= \int _{\varOmega }A(x,\nabla u)\,dx, \\& K(u)= \int _{\varOmega }F(x,u)\,dx, \\& L(u)= \int _{\partial \varOmega }B(x,u)\,dx, \\& I(u)=\varLambda (u)+ \int _{\varOmega }\frac{1}{p(x)} \vert u \vert ^{p(x)} \,dx- \int _{ \varOmega }\frac{h(x)}{p^{*}(x)} \vert u \vert ^{p^{*}(x)} \,dx-K(u)-L(u). \end{aligned}$$

We say that \(u\in W^{1,p(x)}(\varOmega )\) is a weak solution of \(p(x)\)-Laplacian problem (1.1), if, for any \(v\in W^{1,p(x)}( \varOmega )\),

$$\begin{aligned} \bigl\langle I'(u),v \bigr\rangle &= \int _{\varOmega }a(x,\nabla u)\nabla v\,dx+ \int _{\varOmega } \vert u \vert ^{p(x)-2}uv \,dx- \int _{\varOmega }h(x) \vert u \vert ^{p^{*}(x)-2}uv \,dx \\ &\quad {} - \int _{\varOmega }f(x,u)v\,dx- \int _{\partial \varOmega }b(x,u)v\,dS=0. \end{aligned}$$

So next we need only to consider the existence of nontrivial critical points of \(I(u)\).

Lemma 3.1

([16], Lemma 1)

The functional Λ is well-defined on \(W^{1,p(x)}(\varOmega )\), and for all \(u,v\in W^{1,p(x)}\),

$$ \bigl\langle \varLambda '(u),v \bigr\rangle = \int _{\varOmega }a(x,\nabla u)\nabla v\,dx. $$

Lemma 3.2

([5], Lemma 2.9)

Suppose that f satisfies (F1) or (F̃1). Then \(K(u)\) is weakly continuous.

Lemma 3.3

([5], Theorem 2.10)

Suppose that f satisfies (F1) or (F̃1). Then \(K(u)\) is differentiable on \(W^{1,p(x)}\), and, for all \(u,v\in W^{1,p(x)}\),

$$ \bigl\langle K'(u),v \bigr\rangle = \int _{\varOmega }f(x,u)v\,dx. $$

In the same way, the function L leads to a conclusion similar to Lemma 3.2 and Lemma 3.3.

Lemma 3.4

([20], Theorem 4.1)

The mapping a is an operator of type \({S_{+}}\), that is, if \(u_{n}\to u\) weakly in \(W^{1,p(x)}(\varOmega )\) and

$$ \lim_{n\to \infty }\sup \int _{\varOmega }a(x,\nabla u)\cdot ( \nabla u_{n}-\nabla u)\,dx\leq 0, $$
(3.1)

then \(u_{n}\to u\) strongly in \(W^{1,p(x)}(\varOmega )\).

Theorem 3.1

Assume hypotheses (F1), (F3), (F4), (B1), (B3) and (H1) are fulfilled. Then there exists \(M>0\) such that, whenever \(h(x)\leq M\), the problem has a nontrivial solution.

Proof

(1) There exists \(r>0\) such that \(\inf \{I(u):|\!|\!|u|\!|\!|=r, u\in W ^{1,p(x)}(\varOmega )\}>c\).

From (F1), (F4) and (B1) we have

$$\begin{aligned}& \bigl\vert F(x,u) \bigr\vert \leq \varepsilon \vert u \vert ^{p(x)}+C(\varepsilon ) \vert u \vert ^{p^{*}(x)}, \\& \bigl\vert B(x,u) \bigr\vert \leq C_{2}\frac{1}{\alpha _{2}(x)} \vert u \vert ^{\alpha _{2}(x)}. \end{aligned}$$

Next, from (A4),

$$\begin{aligned} I(u) &= \int _{\varOmega }A(x,\nabla u)\,dx+ \int _{\varOmega }\frac{1}{p(x)} \vert u \vert ^{p(x)} \,dx+ \int _{\varOmega }\frac{h(x)}{p^{*}(x)} \vert u \vert ^{p^{*}(x)} \,dx \\ &\quad {} - \int _{\varOmega }F(x,u)\,dx- \int _{\partial \varOmega }B(x,u)\,dx \\ &\geq \int _{\varOmega }\frac{1}{p(x)} \vert \nabla u \vert ^{p(x)}+ \frac{1}{p(x)} \vert u \vert ^{p(x)}- \frac{h(x)}{p ^{*}(x)} \vert u \vert ^{p^{*}(x)}\,dx \\ &\quad {} - \int _{\varOmega } \bigl(\varepsilon \vert u \vert ^{p(x)}+C( \varepsilon ) \vert u \vert ^{p ^{*}(x)} \bigr)\,dx - \int _{\partial \varOmega }\frac{C_{2}}{\alpha _{2}(x)} \vert u \vert ^{ \alpha _{2}(x)} \,dS \\ &\geq \frac{1}{p_{2}} \int _{\varOmega } \vert \nabla u \vert ^{p(x)}\,dx+ \frac{1}{p _{2}} \int _{\varOmega } \vert u \vert ^{p(x)}\,dx- \frac{h_{1}}{p^{*}_{1}} \int _{\varOmega } \vert u \vert ^{p ^{*}(x)}\,dx \\ &\quad {} - \int _{\varOmega }\varepsilon \vert u \vert ^{p(x)}+C( \varepsilon ) \vert u \vert ^{p ^{*}(x)}\,dx - \int _{\partial \varOmega }\frac{C_{2}}{\alpha _{2}(x)} \vert u \vert ^{ \alpha _{2}(x)} \,dS. \end{aligned}$$

Let \(\varepsilon <\frac{1}{2p_{2}}\), we get

$$ I(u)\geq \int _{\varOmega }(\frac{ \vert \nabla u \vert ^{p(x)}+ \vert u \vert ^{p(x)}}{2p_{2}}-C \vert u \vert ^{p ^{*}(x)} \,dx- \int _{\partial \varOmega }\frac{C_{2}}{\alpha _{2}(x)} \vert u \vert ^{ \alpha _{2}(x)} \,dS. $$
(3.2)

As \(\alpha _{2}(x)\), \(p(x)\) are continuous on Ω̅, there exists \(\delta _{1}>0\) such that \(\vert \alpha _{2}(x)-\alpha _{2}(y)\vert <\varepsilon \) and \(\vert p(x)-p(y)\vert <\varepsilon \) for any \(\varepsilon \in (0,1)\) whenever \(\vert x-y\vert <\delta _{1}\). Take \(x\in \overline{\varOmega }\), for any \(y\in B_{\delta _{1}(x)}(x)\cap \overline{\varOmega }\), we have

$$ p(y)< p(x)+\varepsilon $$

and

$$ \alpha _{2}(y)>\alpha _{2}(x)-\varepsilon . $$

As \(p(x)\ll \alpha _{2}(x)\), take \(\varepsilon =\frac{1}{4}\inf_{x \in \overline{\varOmega }}(\alpha _{2}(x)-p(x))\), we have

$$ \alpha _{2}(x)-\varepsilon - \bigl(p(x)+\varepsilon \bigr)\geq \frac{1}{2} \inf_{x\in \overline{\varOmega }} \bigl(\alpha _{2}(x)-p(x) \bigr)>0, $$

then

$$ p(y)< p(x)+\varepsilon < \alpha _{2}(x)-\varepsilon < \alpha _{2}(y), $$

and further

$$ p_{x}^{-}=\sup_{y\in \overline{B_{\delta _{1}}(x)}}p(y)< \alpha _{2x} ^{-}=\inf_{y\in \overline{B_{\delta _{1}}(x)}} \bigl(\alpha _{2}(x) \bigr). $$

In the same manner, we get

$$ \alpha _{2x}^{+}=\sup_{y\in \overline{B_{\delta _{1}}(x)}}\alpha _{2}(x)< p _{x}^{*-}=\inf_{y\in \overline{B_{\delta _{1}}(x)}}p^{*}(x). $$

\(\{B_{\delta _{x}}(x),x\in \overline{\varOmega }\}\) is an open covering of Ω̅. Since Ω̅ is compact, we can pick a finite subcovering \(\{B_{\delta _{i}}(x_{i})\}_{i=1}^{k}\) for Ω̅ from the covering \(\{B_{\delta _{x}}(x),x\in \overline{ \varOmega }\}\) such that \(\bigcup_{i=1}^{k} B_{\delta _{i}}(x_{i})\supset \overline{\varOmega }\). Denote \(\delta _{l}=\min \{ \delta _{i},i=1,2,\ldots,k\}\), we can use all the hypercubes whose length of the side is \(\frac{\delta _{l}}{2}\) to divide the entire space \(\mathbb{R}^{N}\), then \(\bigcup_{i=1}^{k} B_{\delta _{i}}(x_{i})\cap \varOmega \) is divided by finite open regions \(\{\varOmega _{i}\}_{i=1}^{m}\) which mutually have no common points, and \(\overline{\varOmega }=\bigcup_{i=1}^{m}\overline{ \varOmega _{i}}\). Then

$$ p_{i}^{-}=\inf_{x\in \overline{\varOmega _{i}}}< p_{i}^{+}= \sup_{x\in \overline{\varOmega _{i}}}p(x)< \alpha _{2i}^{-}= \inf _{x\in \overline{\varOmega _{i}}}\alpha _{2}(x)< p_{i}^{*-}= \inf_{x\in \overline{\varOmega _{i}}}p^{*}(x). $$
(3.3)

By Theorems 2.5 and 2.6, we know that there exist \(c_{4},c_{5}>1\) such that

$$ \Vert u \Vert _{L^{p^{*}}(\varOmega _{i})}\leq c_{4} |\!|\!|u |\!|\!|_{\varOmega _{i}},\quad\quad \Vert u \Vert _{L^{\alpha _{2}}(\partial \varOmega _{i})}\leq c_{5} |\!|\!|u |\!|\!|_{\varOmega _{i}}, $$

where \(i=1,2,\ldots,m\).

Take \(|\!|\!|u|\!|\!|\leq [\max (c_{4},c_{5})]^{-1}\), then \(|\!|\!|u|\!|\!|_{\varOmega _{i}}<[\max (c_{4},c_{5})]^{-1}\) and

$$ \Vert u \Vert _{L^{p^{*}}(\varOmega _{i})}< 1, \quad\quad \Vert u \Vert _{L^{\alpha _{2}}(\partial \varOmega _{i})}< 1, $$

then we have

$$\begin{aligned} & \int _{\varOmega } \biggl(\frac{ \vert \nabla u \vert ^{p(x)}+ \vert u \vert ^{p(x)}}{2p_{2}}-C \vert u \vert ^{p ^{*}(x)} \biggr)\,dx- \int _{\partial \varOmega }\frac{C_{2}}{\alpha _{2}(x)} \vert u \vert ^{ \alpha _{2}(x)} \,dS \\ &\quad =\sum_{i=1}^{m} \int _{\varOmega _{i}}(\frac{ \vert \nabla u \vert ^{p(x)}+ \vert u \vert ^{p(x)}}{2p _{2}}-C \vert u \vert ^{p^{*}(x)} \,dx-\sum_{i=1}^{m} \int _{\partial \varOmega _{i}}\frac{C _{2}}{\alpha _{2}(x)} \vert u \vert ^{\alpha _{2}(x)} \,dS \\ &\quad \geq \sum_{i=1}^{m} \biggl( \frac{1}{2p_{2}} |\!|\!|u |\!|\!|_{\varOmega _{i}}^{p_{i}^{+}}-C |\!|\!|u |\!|\!|_{ \varOmega _{i}}^{p_{i}^{*-}}-C |\!|\!|u |\!|\!|_{\varOmega _{i}}^{\alpha _{2i}^{-}} \biggr). \end{aligned}$$

Let

$$ g(t)=\frac{1}{2p_{2}}t^{p^{+}_{i}}-Ct^{p^{*-}_{i}}-Ct^{\alpha ^{-}_{2i}}. $$
(3.4)

By (3.3), there exists \(0< t_{i}<1\) such that \(g(t)\) is positive and increasing for any \(t\in (0,t_{i}]\).

Take \(t_{k}=\min \{t_{i},i=1,2,\ldots,m\}\). Since \(|\!|\!|u|\!|\!|\leq \sum_{i=1} ^{m} |\!|\!|u|\!|\!|_{\varOmega _{i}}\), when \(|\!|\!|u|\!|\!|=r< t_{k}\), there exists j such that \(\frac{r}{m}\leq |\!|\!|u|\!|\!|_{\varOmega _{j}}\leq r< t _{j}\), then

$$\begin{aligned} I(u) &\geq \int _{\varOmega } \biggl(\frac{ \vert \nabla u \vert ^{p(x)}+ \vert u \vert ^{p(x)}}{2p_{2}}-C \vert u \vert ^{p ^{*}(x)} \biggr)\,dx- \int _{\partial \varOmega }\frac{C_{2}}{\alpha _{2}(x)} \vert u \vert ^{ \alpha _{2}(x)} \,dS \\ &\geq \frac{1}{2p_{2}} |\!|\!|u |\!|\!|_{\varOmega _{j}}^{p^{+}_{j}}-C |\!|\!|u |\!|\!|_{ \varOmega _{j}}^{p^{*-}_{j}}-C |\!|\!|u |\!|\!|_{\varOmega _{j}}^{\alpha ^{-}_{2j}} \\ &\geq \biggl(\frac{r}{m} \biggr)^{p^{+}_{j}} \biggl[ \frac{1}{2p_{2}}-C \biggl(\frac{r}{m} \biggr)^{p ^{*-}_{j}-p^{+}_{j}}-C \biggl( \frac{r}{m} \biggr)^{\alpha ^{-}_{2j}-p^{+}_{j}} \biggr] \\ &\geq \biggl(\frac{r}{m} \biggr)^{p^{+}_{j}} \biggl[ \frac{1}{2p_{2}}-C \biggl(\frac{r}{m} \biggr)^{ \alpha ^{-}_{2j}-p^{+}_{j}}-C \biggl( \frac{r}{m} \biggr)^{\alpha ^{-}_{2j}-p^{+}_{j}} \biggr]. \end{aligned}$$

Take

$$ r=\min \biggl\{ m \biggl(\frac{1}{4Cp_{2}} \biggr)^{\frac{1}{\alpha _{2j}^{-}-p_{j}^{+}}},t _{k} \biggr\} , $$

we have \(I(u)\geq c\), where \(c=\frac{1}{4p_{2}}(\frac{r}{m})^{p_{j} ^{+}}\).

(2) There exists \(e\in W^{1,p(x)}(\varOmega )\) such that \(|\!|\!|e|\!|\!|>r\), then we have \(I(e)<0\).

From (F1) and (F3), we have

$$ F(x,u)\geq C \vert u \vert ^{\mu _{1}(x)}, $$

for any \((x,t)\in {\varOmega _{0}}\times \mathbb{R}\).

Next from (A1) and (A2), for any \(x\in \overline{\varOmega }\),

$$ A(x,\nabla u)= \int _{0}^{1}a(x,t\nabla u)\,dt\leq c_{0} \biggl( \vert \nabla u \vert + \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)} \biggr) $$

and

$$\begin{aligned} I(u) &\leq \int _{\varOmega }c_{0} \vert \nabla u \vert \,dx+ \int _{\varOmega } \frac{c_{0}}{p(x)} \vert \nabla u \vert ^{p(x)}+\frac{1}{p(x)} \vert u \vert ^{p(x)}\,dx- \int _{\varOmega }\frac{h(x)}{p^{*}(x)} \vert u \vert ^{p^{*}(x)} \\ &\quad {} - \int _{\varOmega }C \vert u \vert ^{\mu _{1}(x)}\,dx- \int _{\partial \varOmega }B(x,u)\,dx. \end{aligned}$$

Pick \(x_{0}\in \varOmega _{0}\). As \(\mu _{1}\), p is continuous on Ω̅, there exists \(0<2R<1\) such that

$$ p_{2x_{0}}=\sup_{x\in B_{2R}(x_{0})}p(x)< \mu _{1x_{0}}^{-}= \inf_{x\in B_{2R}(x_{0})}\mu _{1}(x)\leq \mu _{1x_{0}}^{+}= \sup _{x\in B_{2R}(x_{0})} $$
(3.5)

for \(B_{2R}(x_{0})\subset \varOmega _{0}\). Let \(\phi \in C_{0}^{\infty }(B _{2R}(x_{0}))\) such that \(\phi \equiv 1\) for any \(x\in B_{2R}(x_{0})\), \(0 \leq \phi \leq 1\) and \(\vert \nabla \phi \vert \leq \frac{1}{R}\). Then, for \(s>1\),

$$\begin{aligned} I(s\phi ) &\leq \int _{\varOmega }c_{0}s \vert \nabla \phi \vert \,dx+ \int _{\varOmega }\frac{c _{0}}{p(x)} \vert \nabla s\phi \vert ^{p(x)}+\frac{1}{p(x)} \vert s\phi \vert ^{p(x)}\,dx- \int _{\varOmega }C \vert s\phi \vert ^{\mu _{1}(x)}\,dx \\ &\leq \frac{c_{0}}{R} \int _{B_{2R}(x_{0})}s\,dx+ \int _{B_{2R}(x_{0})}\frac{c _{0}}{p_{1}R^{p_{2x_{0}}}}s^{p(x)} + \frac{1}{p_{1}}s^{p(x)}\,dx \\ &\quad {} - \int _{B_{2R}(x_{0})}Cs\mu _{1x_{0}}(x) \vert \phi \vert ^{\mu _{1}(x)}\,dx \\ &\leq \frac{c_{0}}{R} \int _{B_{2R}(x_{0})}s\,dx+ \int _{B_{2R}(x_{0})} \biggl(\frac{c _{0}}{p_{1}R^{p_{2x_{0}}}}+\frac{1}{p_{1}} \biggr)s^{p(x)}\,dx \\ &\quad {} - \int _{B_{2R}(x_{0})}Cs\mu _{1x_{0}}^{-} \vert \phi \vert ^{\mu _{1}(x)}\,dx \\ &\leq \int _{B_{2R}(x_{0})}s^{p(x)}(\frac{c_{0}}{R}s^{1-p(x)}+ \frac{c _{0}}{p_{1}R^{p_{2x_{0}}}}+\frac{1}{p_{1}}-\overline{C}s^{\mu _{1x_{0}} ^{-}-p(x)}\,dx, \end{aligned}$$

where \(\overline{C}=\frac{C\int _{B_{2R}(x_{0})}\vert \phi \vert ^{\mu _{1}(x)}\,dx}{\vert B _{2R}(x_{0})\vert }\).

As \(\phi \equiv 1\) for any \(x\in B_{2R}(x_{0})\), \(\int _{B_{2R}(x_{0})}\vert \phi \vert ^{\mu _{1}(x)}\,dx>0\), thus \(\overline{C}>0\).

As \(p(x)>1\), if s is sufficiently large, then \(s^{1-p(x)}<1\). Thus

$$\begin{aligned} I(s\phi ) &\leq \int _{B_{2R}(x_{0})}s^{p(x)}(\frac{c_{0}}{R}+ \frac{c _{0}}{p_{1}R^{p_{1}}}+\frac{1}{p_{1}}-\overline{C}s^{\mu _{1}^{-}-p(x)}\,dx \\ &= \int _{B_{2R}(x_{0})}s^{p(x)} \bigl(C-s^{\mu _{1x_{0}}^{-}-p(x)} \bigr)\,dx. \end{aligned}$$

Because \(\mu _{1x_{0}}^{-}-p_{2x_{0}}>0\), when s is sufficiently large, we have \(|\!|\!|s\phi |\!|\!|>r\) and \(I(s\phi )<0\).

(3) The functional I satisfies the (PS) condition (i.e. any sequence \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) with \(I(u_{n})\leq c\) and \(I^{\prime }(u_{n})\rightarrow 0\) as \(i\rightarrow \infty \) in \(W^{-1,p^{\prime }(x)}\) possesses a convergent subsequence).

(i) First, we show that the (PS) sequence \(\{u_{n}\}\subset W^{1,p(x)}\) is bounded.

Note that \(p(x)\) is Lipschitz continuous, then there exists a Lipschitz continuous function \(v(x)\) such that \(p(x)\ll v(x)\leq p^{*}(x)\) and

$$ v_{1}=\inf_{x\in \varOmega }v(x)\leq \sup_{x\in \varOmega }v(x)=v_{2}. $$
(3.6)

Take

$$ v(x)=p(x)+\min \Bigl\{ \inf_{x\in \varOmega } \bigl(\mu _{1}(x)-p(x) \bigr),\inf_{x\in \varOmega } \bigl(\mu _{2}(x)-p(x) \bigr),\inf_{x\in \varOmega } \bigl(p^{*}(x)-p(x) \bigr) \Bigr\} , $$

we obtain

$$\begin{aligned} & I(u_{n})- \biggl\langle I'(u_{n}), \frac{u_{n}}{v(x)} \biggr\rangle \\ &\quad \geq \int _{\varOmega }(\frac{1}{p(x)}a(x,\nabla u_{n}) \nabla u_{n}- \frac{1}{v(x)}a(x,\nabla u_{n})\nabla u_{n}+a(x,\nabla u_{n})\frac{u _{n}}{v(x)^{2}}\nabla v(x) \\ &\quad\quad {} + \int _{\varOmega } \biggl(\frac{1}{p(x)}-\frac{1}{v(x)} \biggr) \vert u_{n} \vert ^{p(x)}\,dx- \int _{\varOmega } \biggl(\frac{1}{p^{*}(x)}-\frac{1}{v(x)} \biggr)h(x) \vert u_{n} \vert ^{p^{*}(x)}\,dx \\ &\quad\quad {} - \int _{\varOmega } \biggl(F(x,u_{n})-\frac{1}{v(x)}f(x,u_{n})u_{n} \biggr)\,dx- \int _{\partial \varOmega } \biggl(B(x,u_{n})-\frac{1}{v(x)}b(x,u_{n})u_{n} \biggr)\,dS \\ &\quad \geq \int _{\varOmega } \biggl(\frac{1}{p(x)}-\frac{1}{v(x)} \biggr) \bigl( \vert \nabla u_{n} \vert ^{p(x)}+ \vert u _{n} \vert ^{p(x)} \bigr)\,dx+ \int _{\varOmega }a(x,\nabla u_{n})\frac{u_{n}}{v(x)^{2}} \nabla v(x)\,dx \\ &\quad \quad {} + \int _{\varOmega } \biggl(\frac{1}{v(x)}-\frac{1}{p^{*}(x)} \biggr)h(x) \vert u_{n} \vert ^{p ^{*}(x)}\,dx \\ &\quad \geq \frac{l_{1}}{v_{2}p_{2}} \int _{\varOmega } \bigl( \vert \nabla u_{n} \vert ^{p(x)}+ \vert u _{n} \vert ^{p(x)} \bigr)\,dx+ \frac{l_{2}h_{1}}{v_{2}p_{2}^{*}} \int _{\varOmega } \vert u_{n} \vert ^{p ^{*}(x)} \,dx \\ &\quad\quad {}- \frac{c_{0}M}{v_{1}^{2}} \int _{\varOmega } \vert u_{n} \vert \bigl(1+ \vert \nabla u _{n} \vert ^{p(x)-1} \bigr)\,dx, \end{aligned}$$

where \(l_{1}=\inf_{x\in \varOmega }\{v(x)-p(x)\}\), \(l_{2}=\inf_{x \in \varOmega }{p^{*}(x)-v(x)\vert }\), \(M= \sup_{x\in \varOmega }\vert \nabla v(x)\vert \).

By the Young inequality, we have

$$ \begin{aligned}[b] \int _{\varOmega } \vert u_{n} \vert \,dx &\leq \int _{\varOmega }\varepsilon _{1} \frac{1}{p(x)} \vert u_{n} \vert ^{p(x)}+\frac{p(x)-1}{p(x)}\varepsilon _{1}^{ \frac{1}{1-p(x)}}\,dx \\ &\leq \frac{\varepsilon _{1}}{p_{1}} \int _{\varOmega } \vert u_{n} \vert ^{p(x)}+C( \varepsilon _{1}). \end{aligned} $$
(3.7)

Take \(\varepsilon _{1}=\min \{1,\frac{v_{1}^{2}p_{1}l_{1}}{2c_{0}Mv _{2}p_{2}}\}\) such that \(\frac{c_{0}M\varepsilon _{1}}{v_{1}^{2}p_{1}} \leq \frac{l_{1}}{2v_{2}p_{2}}\).

By the Young inequality, we have

$$ \int _{\varOmega } \vert \nabla u_{n} \vert ^{p(x)-1} \vert u_{n} \vert \,dx\leq \varepsilon _{2} \int _{\varOmega } \vert \nabla u_{n} \vert ^{p(x)} \,dx+\frac{\varepsilon _{2}^{1-p_{2}}}{p _{1}} \int _{\varOmega } \vert u_{n} \vert ^{p(x)} \,dx. $$

Take \(\varepsilon _{2}=\min \{1,\frac{v_{1}^{2}l_{1}}{2c_{0}Mv_{2}p _{2}}\}\) such that \(\frac{c_{0}M\varepsilon _{2}}{v_{1}^{2}}\leq \frac{l _{1}}{2v_{2}p_{2}}\).

By the Young inequality again, we have

$$\begin{aligned} \int _{\varOmega } \vert u_{n} \vert ^{p(x)} \,dx & \leq \varepsilon _{3} \frac{p(x)}{p^{*}(x)} \vert u_{n} \vert ^{p^{*}(x)}+ \frac{p^{*}(x)-p(x)}{p^{*}(x)} \varepsilon _{3}^{\frac{p(x)}{p(x)-p ^{*}(x)}} \,dx \\ &\leq \frac{p_{2}\varepsilon _{3}}{p_{1}^{*}} \int _{\varOmega } \vert u_{n} \vert ^{p ^{*}(x)} \,dx+C( \varepsilon _{3}). \end{aligned}$$

Take \(\varepsilon _{3}=\min \{1,\frac{h_{1}l_{2}v_{1}^{2}p_{1}p_{1} ^{*}}{2c_{0}Mv_{2}p_{2}^{*}p_{2}\varepsilon _{2}^{1-p_{2}}}\}\) such that \(\frac{c_{0}Mp_{2}\varepsilon _{2}^{1-p_{2}}}{v_{1}^{2}p_{1}p_{1}^{*}} \leq \frac{l_{1}}{2v_{2}p_{2}}\). Then

$$\begin{aligned} & I(u_{n})- \biggl\langle I'(u_{n}), \frac{u_{n}}{v(x)} \biggr\rangle \\ &\quad \geq \frac{l_{1}}{2v_{2}p_{2}} \int _{\varOmega } \bigl( \vert \nabla u_{n} \vert ^{p(x)}+ \vert u _{n} \vert ^{p(x)} \bigr)\,dx+ \frac{l_{2}h_{1}}{2v_{2}p_{2}^{*}} \int _{\varOmega } \vert u _{n} \vert ^{p^{*}(x)} \,dx-C \\ &\quad \geq \frac{l_{1}}{2v_{2}p_{2}} \int _{\varOmega } \bigl( \vert \nabla u_{n} \vert ^{p(x)}+ \vert u _{n} \vert ^{p(x)} \bigr)\,dx-C. \end{aligned}$$

As

$$ \int _{\varOmega } \biggl\vert \frac{u_{n}v_{1}}{v(x) \Vert u_{n} \Vert _{L^{p(x)}}} \biggr\vert ^{p(x)}\,dx\leq \int _{\varOmega } \biggl\vert \frac{u_{n}}{ \Vert u_{n} \Vert _{L^{p(x)}}} \biggr\vert ^{p(x)}\,dx\leq 1, $$

we have \(\Vert \frac{u_{n}}{v}\Vert _{L^{p(x)}}\leq \frac{ \Vert u_{n}\Vert _{L^{p(x)}}}{v_{1}}\). Since

$$ \biggl\Vert u_{n}\nabla \frac{1}{v(x)} \biggr\Vert _{L^{p(x)}}= \biggl\Vert \frac{u _{n}\nabla v(x)}{v^{2}(x)} \biggr\Vert _{L^{p(x)}} \leq \frac{M}{v_{1}^{2}} \Vert u_{n} \Vert _{L^{p(x)}}, $$

we have

$$\begin{aligned} \begin{aligned} \biggl\Vert \nabla \frac{u_{n}}{v(x)} \biggr\Vert _{L^{p(x)}} &= \biggl\Vert u _{n} \nabla \frac{1}{v(x)}+ \frac{\nabla u_{n}}{v(x)} \biggr\Vert _{L^{p(x)}} \\ &\leq \frac{M}{v_{1}^{2}} \Vert u_{n} \Vert _{L^{p(x)}}+ \frac{M+v _{1}}{v_{1}^{2}} \Vert u_{n} \Vert _{W^{1,p(x)}}, \end{aligned} \end{aligned}$$

so

$$\begin{aligned} \biggl\Vert \frac{u_{n}}{v(x)} \biggr\Vert _{W^{1,p(x)}} &\leq \biggl\Vert \frac{u _{n}}{v} \biggr\Vert _{L^{p(x)}}+ \biggl\Vert \nabla \frac{u_{n}}{v(x)} \biggr\Vert _{L^{p(x)}} \\ &\leq \frac{ \Vert u_{n} \Vert _{L^{p(x)}}}{v_{1}}+\frac{M+v _{1}}{v_{1}^{2}} \Vert u_{n} \Vert _{W^{1,p(x)}}\leq C \Vert u_{n} \Vert _{W^{1,p(x)}}, \end{aligned}$$

where C is constant. Moreover, \(\frac{\Vert u_{n} \Vert _{W^{1,p(x)}}}{2}\leq |\!|\!|u|\!|\!|\leq 2\Vert u_{n} \Vert _{W^{1,p(x)}}\), we have

$$ \biggl|\!\biggl|\!\biggl|\frac{u_{n}}{v(x)} \biggr|\!\biggr|\!\biggr|\leq 2 \biggl\Vert \frac{u_{n}}{v} \biggr\Vert _{W^{1,p(x)}}\leq \frac{4C}{ |\!|\!|u |\!|\!|}, $$

when n is sufficiently large, we obtain

$$ C+C |\!|\!|u_{n} |\!|\!|\geq \frac{l_{1}}{2v_{2}p_{2}} \int _{\varOmega } \bigl( \vert \nabla u _{n} \vert ^{p(x)}+ \vert u_{n} \vert ^{p(x)} \bigr)\,dx\geq \frac{l_{1}}{2v_{2}p_{2}} |\!|\!|u_{n} |\!|\!|^{p _{1}}. $$

By the Young inequality, we have

$$ |\!|\!|u_{n} |\!|\!|\leq \frac{\varepsilon }{p_{1}} |\!|\!|u_{n} |\!|\!|^{p_{1}}+C( \varepsilon ). $$

Take \(\varepsilon =\frac{l_{1}p_{1}}{4v_{2}p_{2}C}\) such that

$$ C+\frac{l_{1}}{4v_{2}p_{2}} |\!|\!|u_{n} |\!|\!|^{p_{1}}\geq \frac{l_{1}}{2v _{2}p_{2}} |\!|\!|u_{n} |\!|\!|^{p_{1}}, $$

then \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) is bounded.

(ii) Next, we show that the (PS) sequence \(\{u_{n}\}\subset W^{1,p(x)}( \varOmega )\) possesses a convergent subsequence.

We know that \(\{u_{n}\}\) is bounded. As \(W^{1,p(x)}(\varOmega )\) is reflexive, passing to a subsequence (still denoted by \(\{u_{n}\}\)), we may assume that there exists \(u\in W^{1,p(x)}(\varOmega )\) such that \(u_{n}\rightarrow u\) weakly in \(W^{1,p(x)}\) and \(u_{n}\rightarrow u\) a.e. on Ω.

From the definition of (PS) sequence, we obtain \(\lim_{n\rightarrow \infty }\langle I'(u_{n}),u_{n}-u\rangle =0\), i.e.

$$\begin{aligned} &\lim_{n\rightarrow \infty } \biggl[ \int _{\varOmega }a(x, \nabla u_{n}) (\nabla u_{n}-\nabla u)\,dx+ \int _{\varOmega } \vert u_{n} \vert ^{p(x)-2}u _{n}(u_{n}-u)\,dx \\ & \quad {} - \int _{\varOmega }f(x,u_{n}) (u_{n}-u)\,dx- \int _{\varOmega }h(x) \vert u_{n} \vert ^{p^{*}(x)-2}(u _{n}-u)\,dx \\ & \quad {} - \int _{\partial \varOmega }b(x,u_{n}) (u_{n}-u)\,dx \biggr]=0. \end{aligned}$$

As \(p(x)< p^{*}(x)\), the embedding \(W^{1,p(x)}\rightarrow L^{p(x)}( \varOmega )\) is compact, so \(u_{n}\rightarrow u\) strongly in \(L^{p(x)}( \varOmega )\). Hence when \(n\rightarrow \infty \),

$$ \biggl\vert \int _{\varOmega } \vert u_{n} \vert ^{p(x)-2}u_{n}(u_{n}-u) \,dx \biggr\vert \leq \bigl\Vert \vert u_{n} \vert ^{p(x)-1} \bigr\Vert _{L^{\frac{p(x)}{p(x)-1}}(\varOmega )} \Vert u_{n}-u \Vert _{L^{p(x)}}\rightarrow 0. $$

From Theorems 2.1 and 2.4,

$$ \biggl\vert \int _{\varOmega } f(x,u_{n}) (u_{n}-u) \,dx \biggr\vert \leq \bigl\Vert f(x,u_{n}) \bigr\Vert _{L^{\frac{\alpha _{1}(x)}{\alpha _{1}(x)-1}}(\varOmega )} \Vert u_{n}-u \Vert _{L^{\alpha _{1}(x)}} \rightarrow 0. $$

From Theorems 2.1 and 2.6,

$$ \biggl\vert \int _{\varOmega } b(x,u_{n}) (u_{n}-u) \,dx \biggr\vert \leq 2 \bigl\Vert b(x,u_{n}) \bigr\Vert _{L^{\frac{\alpha _{2}(x)}{\alpha _{2}(x)-1}}(\varOmega )} \Vert u_{n}-u \Vert _{L^{\alpha _{2}(x)}}\rightarrow 0. $$

If we could verify that \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}( \varOmega )\), we can obtain

$$ \int _{\varOmega }h(x) \vert u_{n} \vert ^{p^{*}(x)-2}(u_{n}-u)\,dx\vert \leq 2 \bigl\Vert h(x) \bigr\Vert _{L^{\infty }(\varOmega )} \bigl\Vert \vert u_{n} \vert ^{p^{*}(x)-1} \bigr\Vert _{L^{\frac{p^{*}(x)}{p^{*}(x)-1}}(\varOmega )} \Vert u_{n}-u \Vert _{L^{p^{*}(x)}}\rightarrow 0. $$

Therefore, \(\lim_{n\rightarrow \infty }\int _{\varOmega }a(x, \nabla u_{n})\,dx=0\), by Lemma 3.4, a is a \({S}_{+}\) type operator, then \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}( \varOmega )\). □

Next, in order to complete Theorem 3.1, we prove the following lemma.

Lemma 3.5

Let the assumptions of Theorem 3.1 be satisfied. If the (PS) sequence \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) is bounded, then there exists \(M>0\) such that whenever \(h(x)\leq M\), \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}(\varOmega )\).

Proof

As \(u_{n}\rightarrow u\) strongly in \(L^{p(x)}(\varOmega )\), there exists subsequence (still denoted by \(\{u_{n}\}\)), \(u_{n}\rightarrow u\) a.e. on Ω. Note that \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) is bounded, by Borel measure theory, we may assume that

$$\begin{aligned}& \vert \nabla u_{n} \vert ^{p(x)}\rightarrow \mu \quad \text{weak-$*$ in } M(\bar{ \varOmega }), \\& \vert u_{n} \vert ^{p^{*}(x)}\rightarrow \nu \quad \text{weak-$*$ in } M(\bar{ \varOmega }). \end{aligned}$$

\(M(\overline{\varOmega })\) is the space of finite nonnegative Borel measures on Ω.

From the principle of concentration compactness,

$$\begin{aligned}& \mu = \vert \nabla u \vert ^{p(x)}+\sum_{j\in J} \mu _{j}\delta _{x_{j}}+ \widetilde{\mu }, \\& \nu = \vert u \vert ^{p^{*}(x)}+\sum_{j\in J} \nu _{j}\delta _{x_{j}}, \end{aligned}$$

where J is a countable set, \(\{x_{j},j\in J\}\subset \overline{ \varOmega }\), \(\{\nu _{j}\}\subset [0,+\infty )\), \(\delta _{x_{j}}\) is a measure concentrating upon \(x_{j}\), μ̃ is a nonnegative non-atomic measure.

a. First, we show that \(\mu (\{x_{j}\})=\nu (\{x_{j}\})=0\) for any \(j\in J\).

As Ω̅ is compact, so we only need to verify, for any \(x\in \overline{\varOmega }\), there exists \(r_{0}>0\) such that \(\mu (\{x_{j}\})=\nu (\{x_{j}\})=0\) for \(x_{j}\in \overline{\varOmega } \cap B_{r_{0}}(x)\).

Note that \(p(x)\) is Lipschitz continuous and \(p(x)\ll p^{*}(x)\), there exists \(r_{0}>0\) such that

$$ p_{x}^{+}= \sup_{y\in \overline{\varOmega }\cap B_{r_{0}}(x)} p(y)< p_{x}^{*-}=\inf_{y\in \overline{\varOmega }\cap B_{r _{0}}(x)} p^{*}(y). $$

For any \(\varepsilon >0\), let \(\phi _{\varepsilon }\in C_{0}^{\infty }(B _{2\varepsilon }(x_{j}))\) such that \(\phi \equiv 1\) for any \(x\in B_{2\varepsilon }(x_{j})\), \(0\leq \phi _{\varepsilon }\leq 1\) and \(\vert \nabla \phi _{\varepsilon }\vert \leq \frac{2}{\varepsilon }\). Note that

$$\begin{aligned}& \int _{\varOmega } \vert u_{n}\phi _{\varepsilon } \vert ^{p(x)}\,dx \leq \int _{\varOmega } \vert u _{n} \vert ^{p(x)} \,dx, \\& \begin{aligned} \int _{\varOmega } \bigl\vert \nabla (u _{n}\phi _{\varepsilon }) \bigr\vert ^{p(x)}\,dx &= \int _{\varOmega } \vert \nabla u _{n}\cdot \phi _{\varepsilon }+\nabla \phi _{\varepsilon }\cdot u_{n} \vert ^{p(x)}\,dx \\ &\leq \int _{\varOmega }2^{p_{2}}( \vert \nabla u _{n} \vert ^{p(x)}+ \vert \nabla \phi _{\varepsilon } \vert ^{p(x)} \vert u _{n} \vert ^{p(x)}\,dx. \end{aligned} \end{aligned}$$

Since \(u_{n}\in W^{1,p(x)}(\varOmega )\), \(\{u_{n}\phi _{\varepsilon }\}\) is bounded on \(W^{1,p(x)}(\varOmega )\), we have \(\langle I'(u_{n}),u_{n} \phi _{\varepsilon }\rangle \rightarrow 0\) as \(n\rightarrow \infty \). Note that

$$\begin{aligned} \bigl\langle I'(u_{n}),u_{n}\phi _{\varepsilon } \bigr\rangle &= \int _{\varOmega }a(x, \nabla u_{n})\nabla (u_{n}\phi _{\varepsilon })\,dx + \int _{\varOmega } \vert u _{n} \vert ^{p(x)} \phi _{\varepsilon }\,dx- \int _{\varOmega }h(x) \vert u_{n} \vert ^{p^{*}(x)} \phi _{\varepsilon }\,dx \\ &\quad {} - \int _{\varOmega }f(x,u_{n})u_{n}\phi _{\varepsilon }\,dx- \int _{\partial \varOmega }b(x,u_{n})u_{n}\phi _{\varepsilon }\,dx \\ &\geq \int _{\varOmega }a(x,\nabla u_{n})\nabla \phi _{\varepsilon } \cdot u_{n}\,dx + \int _{\varOmega } \vert \nabla u_{n} \vert ^{p(x)} \phi _{\varepsilon }\,dx+ \int _{\varOmega } \vert u_{n} \vert ^{p(x)} \phi _{\varepsilon }\,dx \\ &\quad {} - \int _{\varOmega }h(x) \vert u_{n} \vert ^{p^{*}(x)}\phi _{\varepsilon }\,dx - \int _{\varOmega }f(x,u_{n})u_{n}\phi _{\varepsilon }\,dx- \int _{\partial \varOmega }b(x,u_{n})u_{n}\phi _{\varepsilon }\,dx. \end{aligned}$$

Since \(\vert f(x,u_{n}) \vert \leqslant C_{1}(1+ \vert u_{n} \vert ^{\alpha _{1}(x)-1})\), \(\vert f(x,u_{n})u_{n} \vert \leqslant C _{1}(1+ \vert u_{n} \vert ^{\alpha _{1}(x)})\). So there exists \(\delta >0\) such that, for \(mE<\delta \),

$$ \int _{E}f(x,u_{n})u_{n}\phi _{\varepsilon }\,dx\leqslant C \bigl\Vert 1+ \vert u_{n} \vert ^{\alpha _{1}(x)} \bigr\Vert _{L^{\frac{p^{*}(x)}{ \alpha _{1}(x)}}(E)} \Vert \phi _{\varepsilon } \Vert _{L^{(\frac{p ^{*}(x)}{\alpha _{1}(x)})^{\prime }}(E)}\rightarrow 0. $$

From the Vitali theorem, \(\int _{\varOmega }f(x,u_{n})u_{n} \phi _{\varepsilon }\,dx\rightarrow \int _{\varOmega }f(x,u)u\phi _{\varepsilon }\,dx\). In the same way, \(\int _{\partial \varOmega }b(x, u_{n})u_{n} \phi _{\varepsilon }\,dx\rightarrow \int _{\partial \varOmega }b(x, u)u \phi _{\varepsilon }\,dx\). Then

$$\begin{aligned}& \begin{aligned} & \lim_{n\rightarrow \infty } \int _{\varOmega }a(x,\nabla u_{n})\cdot u_{n} \cdot \nabla \phi _{\varepsilon }\,dx \\ &\quad \leq - \int _{\varOmega }\phi _{\varepsilon }\,d\mu - \int _{\varOmega } \vert u \vert ^{p(x)}\phi _{\varepsilon }\,d\mu + \int _{\varOmega }h(x)\phi _{\varepsilon }\,d \nu \\ &\quad\quad {} - \int _{\varOmega }f(x,u)u\phi _{\varepsilon }\,dx- \int _{\partial \varOmega }b(x,u)u\phi _{\varepsilon }\,dx, \end{aligned} \\& \begin{aligned} & \lim_{n\rightarrow \infty } \biggl\vert \int _{\varOmega }a(x, \nabla u_{n})\cdot u_{n}\cdot \nabla \phi _{\varepsilon }\,dx \biggr\vert \\ &\quad \leq \lim_{n\rightarrow \infty } \biggl\vert \int _{\varOmega }c _{0} \bigl(1+ \vert \nabla u_{n} \vert ^{p(x)-1} \bigr)\cdot u_{n}\cdot \nabla \phi _{\varepsilon }\,dx \biggr\vert \\ &\quad \leq \lim_{n\rightarrow \infty }c_{0} \int _{\varOmega }u_{n} \nabla \phi _{\varepsilon }\,dx+ \lim_{n\rightarrow \infty } c _{0} \int _{\varOmega } \vert \nabla u_{n} \vert ^{p(x)-1} \vert u_{n} \nabla \phi _{\varepsilon } \vert \,dx. \end{aligned} \end{aligned}$$

Note that \(u_{n}\rightarrow u\) strongly in \(L^{p(x)}(B_{2\varepsilon }(x_{j}))\), thus, as \(n\rightarrow \infty \), \(\Vert \nabla \phi _{\varepsilon }\cdot u_{n} \Vert _{L^{p(x)}}\rightarrow \Vert \nabla \phi _{\varepsilon }\cdot u \Vert _{L^{p(x)}}\). Then

$$\begin{aligned} \lim_{n\rightarrow \infty } \int _{\varOmega } \vert \nabla u _{n} \vert ^{p(x)-1} \vert u_{n}\nabla \phi _{\varepsilon } \vert \,dx &\leq \lim_{n\rightarrow \infty }\sup \int _{\varOmega } \vert \nabla u _{n} \vert ^{p(x)-1} \vert u_{n}\nabla \phi _{\varepsilon } \vert \,dx \\ &\leq 2\lim_{n\rightarrow \infty }\sup \bigl\Vert \vert \nabla u _{n} \vert ^{p(x)-1} \bigr\Vert _{L^{p^{\prime }(x)}} |\!|\!|u_{n} \nabla \phi _{\varepsilon } |\!|\!|_{L^{p(x)}} \\ &\leq C |\!|\!|u\nabla \phi _{\varepsilon } |\!|\!|_{L^{p(x)}}. \end{aligned}$$

Note that

$$ \int _{\varOmega } \vert \nabla \phi _{\varepsilon }u \vert ^{p(x)}\,dx \leq 2 \bigl\Vert \vert \nabla \phi _{\varepsilon } \vert ^{p(x)} \bigr\Vert _{(\frac{p^{*}(x)}{p(x)})^{\prime },\varOmega \cap B_{2\varepsilon }(x_{j})} \bigl\Vert \vert u \vert ^{p(x)} \bigr\Vert _{\frac{p^{*}(x)}{p(x)}, \varOmega \cap B_{2\varepsilon }(x_{j})} $$

and

$$ \int _{B_{2\varepsilon }(x_{j})} \bigl( \vert \nabla \phi _{\varepsilon } \vert ^{p(x)} \bigr)^{(\frac{p^{*}(x)}{p(x)})^{\prime }}\,dx = \int _{B_{2\varepsilon }(x _{j})} \vert \nabla \phi _{\varepsilon } \vert ^{N}\,dx\leq 2^{2N}\omega _{N}. $$

From absolute continuity of the integral, we have \(\int _{B_{2\varepsilon }(x_{j})\cap \varOmega }( \vert u \vert ^{p(x)})^{\frac{p ^{*}(x)}{p(x)}}\,dx\rightarrow 0\), then \(|\!|\!|u\cdot \nabla \phi _{\varepsilon } |\!|\!|_{L^{p(x)}}\rightarrow 0\) as \(\varepsilon \rightarrow 0\). Therefore

$$ \biggl\vert \int _{B_{2\varepsilon }(x_{j})\cap \varOmega }\nabla \phi _{\varepsilon }\cdot u_{n} \biggr\vert \,dx\rightarrow 0. $$

Similarly, we can also obtain

$$\begin{aligned}& \biggl\vert \int _{\varOmega }f(x,u)u\phi _{\varepsilon }\,dx \biggr\vert \leq \int _{B_{2\varepsilon }(x_{j})\cap \varOmega } \bigl\vert f(x,u)u \bigr\vert \,dx \rightarrow 0, \\& \biggl\vert \int _{\partial \varOmega }b(x,u)u\phi _{\varepsilon }\,dx \biggr\vert \leq \int _{B_{2\varepsilon }(x_{j})\cap \varOmega } \bigl\vert b(x,u)u \bigr\vert \,dx \rightarrow 0, \\& \biggl\vert \int _{\varOmega } \vert u \vert ^{p(x)}\phi _{\varepsilon }\,dx \biggr\vert \leq \int _{B_{2\varepsilon }(x_{j})\cap \varOmega } \bigl\vert \vert u \vert ^{p(x)} \bigr\vert \,dx\rightarrow 0. \end{aligned}$$

Thus

$$ 0\leq -\mu \bigl(\{x_{j}\} \bigr)+h(x_{j})\nu (x_{j}). $$

Similarly, by the principle of concentration compactness

$$ \nu _{j}\leq C_{*}\max \bigl\{ \mu _{j}^{\frac{p_{x}^{*+}}{p_{x}^{-}}}, \mu _{j}^{\frac{p_{x}^{*-}}{p_{x}^{+}}} \bigr\} . $$

Denote \(h_{2}=\sup_{x\in \varOmega } h(x)\). For any \(j\in J\), we have \(\mu _{j}\leq h_{2}\nu _{j}\). Suppose there exists \(j_{0}\in J\) such that \(\mu _{j_{0}}=\mu _{x_{j_{0}}}>0\). If \(\mu _{j_{0}} \geq 1\), then \(\nu _{j_{0}}\leq C_{*}(h_{2}\nu _{j})^{\frac{p_{x}^{*+}}{p_{x}^{-}}}\), and further

$$ \nu _{j_{0}}\geq \bigl[C_{*}^{-1}h_{2}^{-\frac{p_{x}^{*+}}{p_{x}^{-}}} \bigr] ^{\frac{p_{x}^{-}}{p_{x}^{*+}-p_{x}^{-}}}. $$

If \(\mu _{j_{0}}< 1\), then

$$ \nu _{j_{0}}\geq \bigl[C_{*}^{-1}h_{2}^{\frac{p_{x}^{*-}}{p_{x}^{+}}} \bigr] ^{\frac{p_{x}^{+}}{p_{x}^{*}-p_{x}^{+}}}. $$

Note that \(\int _{\varOmega }\vert u_{n}\vert ^{p^{*}(x)}\,dx\) is bounded and \(\int _{\varOmega }\vert u_{n}\vert ^{p^{*}(x)}\,dx\rightarrow \int _{\varOmega }1\,d\nu = \nu (\overline{\varOmega })\) as \(n\rightarrow \infty \), so \(\nu _{j_{0}}= \nu (\{x_{j_{0}}\})\leq \nu (\overline{\varOmega })<\infty \). Since \(p_{x}^{-}\leq p_{x}^{+} < p_{x}^{*-}\leq p_{x}^{*+}\), there exists \(M>0\) such that, for \(h_{2}\leq M\),

$$\begin{aligned}& \nu (\overline{\varOmega })< \bigl[C_{*}^{-1}h_{2}^{-\frac{p_{x}^{*-}}{p _{x}^{+}}} \bigr]^{\frac{p_{x}^{+}}{p_{x}^{*+}-p_{x}^{-}}}, \\& \nu (\overline{\varOmega })< \bigl[C_{*}^{-1}h_{2}^{-\frac{p_{x}^{*+}}{p _{x}^{-}}} \bigr]^{\frac{p_{x}^{-}}{p_{x}^{*+}-p_{x}^{-}}}, \end{aligned}$$

which is a contradiction. So there exists \(M>0\) such that, for \(h(x)\leq M\), \(\nu _{j}=0\), \(\mu _{j}=0\), where any \(j\in J\).

b. Next, we show that \(u_{n}\rightarrow u\) strongly in \(L^{p^{*(x)}}( \varOmega )\) as \(n\rightarrow \infty \). From the discussion above, we know if \(h(x)\leq M\), then \(\nu =\vert u\vert ^{p^{*}(x)}\). Thus

$$ \int _{\varOmega } \vert u_{n} \vert ^{p^{*}(x)} \,dx \rightarrow \int _{\varOmega }1\,d\nu = \int _{\varOmega } \vert u \vert ^{p^{*}(x)}\,dx. $$

As \(\vert u_{n}-u\vert ^{p^{*(x)}}\leq 2^{p_{2}^{*}}(\vert u_{n}\vert ^{p^{(x)}}+\vert u\vert ^{p ^{*(x)}})\), by the Fatou lemma, we have

$$ \begin{aligned} \int _{\varOmega }2^{p^{*}_{2}+1} \vert u \vert ^{p^{*}(x)} \,dx &= \int _{\varOmega }\lim_{n \rightarrow \infty }\inf \bigl(2^{p_{2}^{*}} \vert u_{n} \vert ^{p^{*}(x)} +2^{p_{2}^{*}} \vert u_{n} \vert ^{p^{*}(x)}- \vert u_{n}-u \vert ^{p^{*(x)}} \bigr)\,dx \\ &\leq \lim_{n\rightarrow \infty }\inf \int _{\varOmega } \bigl(2^{p_{2}^{*}} \vert u_{n} \vert ^{p^{*}(x)} +2^{p_{2}^{*}} \vert u_{n} \vert ^{p^{*}(x)}- \vert u _{n}-u \vert ^{p^{*(x)}} \bigr) \\ &\leq 2^{p^{*}_{2}+1} \int _{\varOmega } \vert u \vert ^{p^{*}(x)}\,dx- \lim _{n \rightarrow \infty }\sup \int _{\varOmega } \vert u_{n}-u \vert ^{p^{*}(x)} \,dx, \end{aligned} $$

then \(\lim_{n\rightarrow \infty }\sup \int _{\varOmega }\vert u_{n}-u\vert ^{p ^{*}(x)}\,dx=0\), and further \(\int _{\varOmega }\vert u_{n}-u\vert ^{p^{*}(x)}\,dx \rightarrow 0\). So \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}( \varOmega )\) as \(n\rightarrow \infty \). □

Multiple solutions for the problems

First, let us introduce some notation. Let \(O(N)\) be the group of orthogonal linear transformations in \(\mathbb{R}^{N}\), and G be a subgroup of \(O(N)\). For \(x\neq 0\), we denote the cardinality of \(G_{x}=\{gx:g\in G\} \) by \(\vert G_{x}\vert \) and set \(\vert G\vert =\inf_{x\in \bar{ \varOmega ,x\neq 0}}\vert G_{x}\vert \). An open subset Ω of \(\mathbb{R}^{N}\) is G-invariant if \(g\varOmega =\varOmega \) for any \(g\in G\).

Definition 4.1

Let Ω be a G-invariant open subset of \(\mathbb{R}^{N}\). The action of G on \(W^{1,p(x)}(\varOmega )\) is defined \(gu(x)=u(g^{-1}x)\) for any \(u\in W^{1,p(x)}(\varOmega )\). The subspace of invariant functions is defined by

$$ W^{1,p(x)}_{G}(\varOmega )= \bigl\{ u\in W^{1,p(x)}( \varOmega ):gu=u, \text{for any } g\in G \bigr\} . $$

A functional \(\varphi : W^{1,p(x)}(\varOmega )\rightarrow \mathbb{R}\) is G-invariant if \(\varphi \circ g=\varphi \) for any \(g\in G\).

If the space X is a separable and reflexive Banach space, there exist \(\{e_{n}\}_{n=1}^{\infty }\subset X\) and \(\{f_{n=1}^{\infty }\}\subset X^{*}\) such that

$$ f_{n}(e_{m})=\delta _{n,m}= \textstyle\begin{cases} 1 &\text{if $ n=m $}, \\ 0 &\text{if $ n\neq m $,} \end{cases} $$

and

$$ X=\overline{\operatorname{span}}\{e_{n}:n=1,2,\ldots \},\quad\quad X^{*}=\overline{\operatorname{span}}\{f_{n}:n=1,2,\ldots \}. $$

For \(k=1,2,\ldots \) we denote

$$ X_{k}=\operatorname{span}\{e_{k}\}, \quad\quad Y_{k}= \bigoplus_{j=1} ^{k} X_{j},\quad\quad Z_{k}=\overline{ \bigoplus _{j=1} ^{\infty }X_{j}}. $$

In order to obtain the multiple solutions for the equation, we need the following hypotheses.

Let Ω be a G-invariant subset of \(\mathbb{R}^{N}\), \(p(x)\) is Lipschitz continuous and G-invariant, and it satisfies \(1< p_{1}\leq p(x) \leq p_{2}< N\). We have:

(F5):

\(f(gx,t)=f(x,t)\) for any \(g\in G\), \(x\in \varOmega \), \(t\in \mathbb{R}\).

(B5):

\(b(gx,t)=b(x,t)\) for any \(g\in G\), \(x\in \varOmega \), \(t\in \mathbb{R}\).

(A6):

\(A(x,\nabla gu)=A(x,\nabla u)\) for any \(g\in G\), \(x\in \varOmega \).

In the following, denote \(G=O(N)\). It is immediate that \(I(u)\in C ^{1}(X,\mathbb{R})\) is G-invariant. Then, by the principle of symmetric criticality, we know that u is a critical point of I if and only if u is a critical point of \(I\vert _{W_{G}^{1,p(x)}}\). Therefore, it suffices to prove the existence of a sequence of critical points for I on \(W_{G}^{1,p(x)}\).

In the following, we prove the existence of a sequence of critical points for I by the fountain theorem, and we take \(X=W^{1,p(x)}_{G}( {\varOmega })\).

Lemma 4.1

([21], Lemma 3.3)

For any \(x\in \bar{\varOmega }\), denote \(\psi _{k}=\sup_{u\in Z_{k},|\!|\!|u|\!|\!|=1}\int _{\varOmega }\vert u\vert ^{p ^{*}(x)}\,dx\), then \(\lim_{k\rightarrow \infty }\psi _{k}=0\).

Lemma 4.2

If \(\alpha (x)\in C(\bar{\varOmega })\), \(\alpha (x)>1\) and \(\alpha (x) \ll p_{*}(x)\) for any \(x\in \bar{\varOmega }\), denote \(\gamma _{k}=\sup_{u \in Z_{k},|\!|\!|u|\!|\!|=1}\int _{\partial \varOmega }\vert u\vert ^{\alpha (x)}\,dx\), then \(\lim_{k\rightarrow \infty } \gamma _{k}=0\).

Proof

Because \(0<\gamma _{k+1}\leq {\gamma _{k}}\), \(\gamma _{k}\rightarrow \gamma \geq 0\), there exists \(u_{k}\in Z_{k}\) such that \(|\!|\!|u_{k}|\!|\!|=1\) and

$$ 0\leq \gamma _{k}- \int _{\varOmega } \vert u_{k} \vert ^{p^{*}(x)} \,dx< \frac{1}{k}. $$

As \(W^{1,p(x)}_{G}{(\varOmega )}\) is reflexive, passing to a subsequence (still denoted by \(\{u_{k}\}\)), we may assume that there exists \(u\in W^{1,p(x)}_{G}{(\varOmega )}\) such that \(u_{k}\rightarrow u\) weakly in \(W^{1,p(x)}_{G}{(\varOmega )}\). For any \(f_{m}\in \{f_{n},n=1,2,\ldots \}\), we have \(f_{m}(u_{k})=0\) when \(m< k\), then \(\lim_{k \rightarrow \infty } f_{m}(u_{k})=f_{m}(u)=0\). So for any \(m\in N\), \(f_{m}(u)=0\), which implies that \(u=0\), and further \(u_{k}\rightarrow 0\) weakly in \(W^{1,p(x)}_{G}{(\varOmega )}\). According to Theorem 2.6, the embedding \(W^{1,p(x)}(\varOmega )\rightarrow L^{\alpha (x)}(\partial \varOmega )\) is compact, so \(u_{k}\rightarrow 0\) strongly in \(L^{\alpha (x)}{(\partial \varOmega )}\), that is, \(\Vert u_{k}\Vert _{L^{\alpha (x)}(\partial \varOmega )}\to 0\). Thus \(\gamma _{k}\to 0\) as \(k\to \infty \). □

Theorem 4.1

Assume hypotheses (F1), (F2), (F3) and (F5) or (F̃1), (F2), (F5), (B̃1), (B2), (B3), (B5) and (H1) are fulfilled, \(p(x)\) is a Lipschitz continuous function on Ω̄ and G-invariant. Then there exists \(M>0\) such that, whenever \(h(x)\leq M\), the problem has a sequence of weak solutions \(\{u_{n}\}\) such that \(I(u_{n})\to \infty \), as \(n\to \infty \).

The theorem will be verified by the fountain theorem in three steps.

Proof

(1) For every \(k \in N\), there exists \(\gamma _{k}>0\), such that \(\inf_{u \in Z_{k} ,|\!|\!|u |\!|\!|= \gamma _{k} } I(u) \to \infty \) as \(k \to \infty \).

From (F̃1) and (B̃1)

$$\begin{aligned}& \begin{gathered} \bigl\vert {F(x,u)} \bigr\vert \le {{{C}}}_{1} \biggl( \vert u \vert + \frac{1}{{\alpha _{1} (x)}} \vert u \vert ^{\alpha _{1} (x)} \biggr), \\ \bigl\vert {B(x,u)} \bigr\vert \le {{{C}}}_{2} \biggl( \vert u \vert + \frac{1}{{\alpha _{2} (x)}} \vert u \vert ^{\alpha _{2} (x)} \biggr), \end{gathered} \end{aligned}$$
(4.1)

then

$$\begin{aligned} I(u) &\ge \frac{1}{{p_{2} }} \int _{\varOmega }{ \vert {\nabla u} \vert ^{p(x)} + \vert u \vert ^{p(x)} } \,dx - \int _{\varOmega }{{{ {C}}}_{1} \biggl( \vert u \vert + \frac{1}{{\alpha _{1} (x)}} \vert u \vert ^{\alpha _{1} (x)} \biggr)} \,dx \\ &\quad {} - \int _{\partial \varOmega } {{{{C}}}_{2} \biggl( \vert u \vert + \frac{1}{ {\alpha _{2} (x)}} \vert u \vert ^{\alpha _{2} (x)} \biggr)} \,dx - \int _{\varOmega }{\frac{{h(x)}}{{p^{*} (x)}}} \vert u \vert ^{p^{*} (x)} \,dx. \end{aligned}$$

By the Young inequalities

$$\begin{aligned}& \int _{\varOmega }{ \vert u \vert } \,dx \le \int _{\varOmega }{\frac{1}{ {p^{*} (x)}} \vert u \vert ^{p^{*} (x)} } + \frac{{p^{*} (x) - 1}}{ {p^{*} (x)}}\,dx \le \int _{\varOmega }{\frac{1}{{p^{*} (x)}} \vert u \vert ^{p^{*} (x)} \,dx} + C, \\& \int _{\varOmega }{ \vert u \vert ^{\alpha _{1} (x)} } \,dx \le \int _{ \varOmega }{\frac{{\alpha _{1} (x)}}{{p^{*} (x)}} \vert u \vert ^{p ^{*} (x)} } + \frac{{p^{*} (x) - \alpha _{1} (x)}}{{p^{*} (x)}}\,dx \le \int _{\varOmega }{\frac{{\alpha _{1} (x)}}{{p^{*} (x)}} \vert u \vert ^{p^{*} (x)} } \,dx + C, \\& \int _{\partial \varOmega } { \vert u \vert } \,dx \le \int _{\partial \varOmega } {\frac{1}{{\alpha _{2} (x)}} \vert u \vert ^{\alpha _{2} (x)} } + \frac{{\alpha _{2} (x) - 1}}{{\alpha _{2} (x)}}\,dx \le \int _{\partial \varOmega } {\frac{1}{{\alpha _{2} (x)}} \vert u \vert ^{\alpha _{2} (x)} \,dx} + C, \end{aligned}$$

then

$$ I(u) \ge \frac{1}{{p_{2} }} \int _{\varOmega }{ \vert {\nabla u} \vert ^{p(x)} + \vert u \vert ^{p(x)} } \,dx - C \int _{\varOmega }{ \vert u \vert ^{p^{*} (x)} } \,dx - C \int _{\partial \varOmega } { \vert u \vert ^{ \alpha _{2} (x)} } \,dx - C. $$

Denote

$$ \beta _{k} = \sup_{u \in Z_{k} , |\!|\!|u |\!|\!|= 1} \int _{\varOmega }{ \vert u \vert ^{p^{*} (x)} } \,dx,\quad \quad \omega _{k} = \sup_{u \in Z_{k} , |\!|\!|u |\!|\!|= 1} \int _{\partial \varOmega } { \vert u \vert ^{\alpha _{2} (x)} } \,dx. $$

As \(\alpha _{2} (x) \ll p_{*} (x) \), so by Lemma 4.2 and Lemma 4.1, we obtain \(\omega _{k} \to 0\) and \(\beta _{k} \to 0\) as \(k \to \infty \). Take

$$ \alpha _{2}^{+} = \sup_{x \in \varOmega } \alpha _{2} (x) ,\quad\quad p_{2}^{*} = \sup _{x \in \varOmega } p^{*} (x). $$

Then, for \(|\!|\!|u |\!|\!|> 1\), we have

$$ I(u) \ge \frac{1}{{p_{2} }} |\!|\!|u |\!|\!|^{p_{1} } - C |\!|\!|u |\!|\!|^{p_{2}^{*} } \beta _{k} - C |\!|\!|u |\!|\!|^{\alpha _{2}^{+} } \omega _{k} - C. $$

From the Young inequality

$$ |\!|\!|u |\!|\!|^{\alpha _{2}^{+} } \le \frac{{\alpha _{2}^{+} }}{{p_{2}^{*} }} |\!|\!|u |\!|\!|^{p_{2}^{*} } + C, $$

then

$$ I(u) \ge \frac{1}{{p_{2} }} |\!|\!|u |\!|\!|^{p_{1} } - C(\beta _{k} + \omega _{k} ) |\!|\!|u |\!|\!|^{p_{2}^{*} } - C. $$

Next, we consider the following equation:

$$\begin{aligned} \frac{1}{{2p_{2} }}t^{p_{1} } - C(\beta _{k} + \omega _{k} )t^{p_{2} ^{*} } = 0. \end{aligned}$$
(4.2)

Let \(t_{k}\) be the solution of (4.2),

$$ t_{k} = \biggl[\frac{1}{{2p_{2} C(\beta _{k} + \omega _{k} )}} \biggr]^{\frac{1}{ {p_{2} ^{*} - p_{1} }}}, $$

\(t_{k} \to \infty \) as \(k \to \infty \). We choose \(\gamma _{k} = t_{k}\), thus, for \(|\!|\!|u |\!|\!|= \gamma _{k}\), \(k \to \infty \), we have

$$ I(u) \ge \frac{1}{{2p_{2} }}\gamma _{k} ^{p_{1} } - C \to \infty . $$

(2) For all \(k \in N\), there exists \(\rho _{k} > \gamma _{k}\) such that \(\max_{u \in Y_{k} , |\!|\!|u|\!|\!|= \rho _{k} } I(u) \le 0\) as \(k \to \infty \), where \(\gamma _{k}\) is given by (1).

From (4.1), we have

$$\begin{aligned} I(u) &\leq \int _{\varOmega }c_{0} \vert \nabla u \vert \,dx + \int _{ \varOmega }\frac{c_{0}}{p(x)} \vert \nabla u \vert ^{p(x)} + \frac{1}{p(x)} \vert u \vert ^{p(x)} \,dx - \int _{\varOmega }\frac{h(x)}{p ^{*} (x)} \vert u \vert ^{p^{*} (x)} \,dx \\ &\quad {} - \int _{\varOmega }F(x,u)\,dx - \int _{\partial \varOmega }B(x,u)\,dS \\ &\leq \int _{\varOmega }c_{0} \vert \nabla u \vert \,dx + \int _{\varOmega }\frac{c_{0}}{p_{1}} \vert \nabla u \vert ^{p(x)} + \frac{1}{p_{1}} \vert u \vert ^{p(x)} \,dx - \int _{\varOmega }{\frac{{h_{1} }}{{p_{2} ^{*} }}} \vert u \vert ^{p^{*} (x)} \,dx \\ &\quad {} + \int _{\varOmega }C_{1} \biggl( \vert u \vert + \frac{1}{\alpha _{1} (x)} \vert u \vert ^{\alpha _{1} (x)} \biggr)\,dx. \end{aligned}$$

By the Young inequality

$$\begin{aligned}& \int _{\varOmega }{ \vert u \vert \,dx} \le \int _{\varOmega }{\frac{{\varepsilon _{1} }}{ {p^{*} (x)}} \vert u \vert ^{p^{*} (x)} + \frac{{p^{*} (x) - 1}}{ {p^{*} (x)}}\varepsilon _{1} ^{\frac{1}{{1 - p^{*} (x)}}} \,dx \le } \varepsilon _{1} \int _{\varOmega }{ \vert u \vert ^{p^{*} (x)} \,dx + C( \varepsilon _{1} )}, \\& \begin{aligned} \int _{\varOmega }{ \vert u \vert ^{\alpha _{1} (x)} \,dx} &\le \int _{ \varOmega }{\varepsilon _{2} \frac{{\alpha _{1} (x)}}{{p^{*} (x)}} \vert u \vert ^{p^{*} (x)} + \frac{{p^{*} (x) - \alpha _{1} (x)}}{{p^{*} (x)}}\varepsilon _{1} ^{\frac{1}{ {\alpha _{1} (x) - p^{*} (x)}}} \,dx} \\ &\le \varepsilon _{2} \int _{\varOmega }{ \vert u \vert ^{p^{*} (x)} \,dx + C( \varepsilon _{2} )}. \end{aligned} \end{aligned}$$

We choose \(\varepsilon _{1} = \min \{ 1,\frac{h_{1} }{4C_{1} p _{2}^{*} } \}\), \(\varepsilon _{2} = \min \{ 1,\frac{\alpha _{1}^{-} h_{1}}{4C_{1} p_{2}^{*} } \}\), then \(C_{1} \varepsilon _{1} \le \frac{h_{1} }{4p_{2}^{*} }\), \(\frac{C_{1} \varepsilon _{2} }{ \alpha _{1}^{-} } \le \frac{h_{1} }{4p_{2}^{*} }\). Thus

$$ I(u) \le \int _{\varOmega }{c_{0} \vert {\nabla u} \vert \,dx} + \frac{ {\max \{ c_{0} ,1\} }}{{p_{1} }} \int _{\varOmega }{ \vert {\nabla u} \vert ^{p(x)} + } \vert u \vert ^{p(x)} \,dx - \frac{{h_{1} }}{{2p_{2} ^{*} }} \int _{\varOmega }{ \vert u \vert ^{p^{*} (x)} } \,dx + C. $$

As \(p(x)\), \(p^{*}(x)\) are continuous on Ω̄, and \(p(x) \ll p^{*} (x)\). Similarly to Theorem 3.1 we can get hypercubes \(\{\varOmega _{i}\}_{i=1}^{m}\) which mutually have no common points and \(\overline{\varOmega }= \bigcup_{i = 1}^{m} {\overline{\varOmega _{i} } }\). On \(\varOmega _{i}\),

$$\begin{aligned} p_{i}^{+} = \sup_{x \in \bar{\varOmega }_{i} } p(x) < p _{i}^{* - } = \inf_{x \in \bar{\varOmega }_{i} } p^{*} (x), \end{aligned}$$
(4.3)

then

$$\begin{aligned} I(u) &\le \sum_{i = 1}^{m} { \int _{\varOmega _{i} } {c_{0} \vert {\nabla u} \vert \,dx} } + \frac{{\max \{ c_{0} ,1\} }}{{p_{1} }} \sum_{i = 1}^{m} { \int _{\varOmega _{i} } { \vert {\nabla u} \vert ^{p(x)} + } \vert u \vert ^{p(x)} \,dx} \\ &\quad {} - \frac{{h_{1} }}{{2p_{2}^{*} }}\sum_{i = 1}^{m} { \int _{ \varOmega _{i} } { \vert u \vert ^{p^{*} (x)} } \,dx} + C. \end{aligned}$$

Since \(p(x) > 1\), from the continuous embedding \(L ^{p(x)} (\varOmega ) \to L ^{1} (\varOmega )\), there exists \(C>0\) such that

$$ \Vert {\nabla u} \Vert _{L ^{1} (\varOmega _{i} )} \le C \Vert {\nabla u} \Vert _{L ^{p(x)} (\varOmega _{i} )} \le 2C |\!|\!|u |\!|\!|_{\varOmega _{i} }. $$

Because \(Y_{k}\) is a finite dimensional space, \(|\!|\!|u |\!|\!|\) and \(\Vert u \Vert _{L^{p^{*} (x)} }\) are equivalent. Thus, for any \(i \in \{ 1,2,\ldots,m\}\), \(|\!|\!|u |\!|\!|_{\varOmega _{i} } \ge 1\),

$$ I(u) \le \sum_{i = 1}^{m} { \biggl(2c_{0}C |\!|\!|u |\!|\!|_{\varOmega _{i} } + \frac{{\max \{ c_{0} ,1\} }}{{p_{1} }} |\!|\!|u |\!|\!|_{\varOmega _{i} }^{p_{i}^{+} } - \frac{ {h_{1} }}{{2p_{2}^{*} }} |\!|\!|u |\!|\!|_{ \varOmega _{i} }^{p_{i}^{* - } } \biggr)} + C. $$

Let

$$ g_{i} (t) = 2c_{0}Ct - \frac{{\max \{ c_{0} ,1\} }}{{p_{1} }}t^{p_{i} ^{+} } - \frac{{h_{1} }}{{2p_{2}^{*} }}t^{p_{i}^{* - } }. $$

Due to (4.3), there exist \(M_{i}>0\), \(g_{i} (t)\) negative and monotone decreasing for any \(t\in [M_{i}, +\infty )\), and \(g_{i}(t) \to -\infty \) as \(t\to \infty \). Denote \(t_{0} =\max \{1,M_{i}, i=1,2,\ldots,m\}\), when \(t>t_{0}\), we have \(g_{j} (t)\leq 0\) for \(j\in \{i=1,2,\ldots,m\}\).

For any \(i\in \{1,2,\ldots,m\}\), \(|\!|\!|u|\!|\!|_{\varOmega _{i}}\geq t_{0}\) when \(|\!|\!|u|\!|\!|_{\varOmega _{i}}\) sufficiently large. It is easy to check that \(I(u)\leq 0\). So when \(|\!|\!|u|\!|\!|\) is large enough, we can find that \(|\!|\!|u|\!|\!|_{\varOmega _{i}}\) is sufficiently large for any \(i\in \{1,2,\ldots,m\}\). Thus \(I(u)\leq 0\) when \(|\!|\!|u|\!|\!|=\rho _{k}>\gamma _{k}\).

(3) The functional I satisfies the (PS) condition.

If the function \(f(x)\) satisfies (F1), (F2), (F3) and (F5), the proof is similar to (3) of Theorem 3.1. We only need to change the space \(W^{1,p(x)}\) to \(W^{1,p(x)}_{G}\).

If the function \(f(x)\) satisfies (F̃1), (F2) and (F5), we choose

$$ v(x) = p(x) + \min \Bigl\{ { \inf_{x \in \varOmega } \bigl( \mu _{2} (x) - p(x) \bigr), \inf_{x \in \varOmega } \bigl(p^{*} (x) - p(x) \bigr)} \Bigr\} $$

in the proof of (3) of Theorem 3.1. Then

$$\begin{aligned} & I(u_{n} ) - \biggl\langle {I'(u_{n} ), \frac{{u_{n} }}{{v(x)}}} \biggr\rangle \\ &\quad \ge \int _{\varOmega }{ \biggl(\frac{1}{{p(x)}}a(x,\nabla u_{n} )\nabla u_{n} - \frac{1}{{v(x)}}a(x,\nabla u_{n} )\nabla u_{n} + a(x,\nabla u_{n} ) \frac{ {u_{n} }}{{v(x)^{2} }}\nabla v(x) \biggr)\,dx} \\ &\quad\quad {} + \int _{\varOmega }{ \biggl(\frac{1}{{p(x)}}} - \frac{1}{{v(x)}} \biggr) \vert {u_{n} } \vert ^{p(x)} \,dx - \int _{\varOmega }{ \biggl(\frac{1}{{p^{*} (x)}} - } \frac{1}{{v(x)}} \biggr)h(x) \vert {u_{n} } \vert ^{p^{*} (x)} \,dx \\ &\quad \quad {} - \int _{\varOmega }{ \biggl(F(x,u_{n} ) - \frac{1}{{v(x)}}f(x,u_{n} )u _{n} \biggr)\,dx} \\ &\quad \ge \frac{{l_{1} }}{{2v_{2} p_{2} }} \int _{\varOmega }{ \vert {\nabla u_{n} } \vert ^{p(x)} + \vert {u_{n} } \vert ^{p(x)} } \,dx - \int _{\varOmega }{ \biggl(F(x,u_{n} ) - \frac{1}{{v(x)}}f(x,u_{n} )u_{n} \biggr)\,dx - } C. \end{aligned}$$

From (4.1) and (F̃1), we obtain

$$\begin{aligned} & I(u_{n} ) - \biggl\langle {I'(u_{n} ), \frac{{u_{n} }}{{v(x)}}} \biggr\rangle \\ &\quad \ge \frac{{l_{1} }}{{2v_{2} p_{2} }} \int _{\varOmega }{ \vert {\nabla u_{n} } \vert ^{p(x)} + \vert {u_{n} } \vert ^{p(x)} } \,dx - \int _{\varOmega }{{{{C}}}_{1} \biggl( \vert {u_{n} } \vert + \frac{1}{ {\alpha _{1} (x)}} \vert {u_{n} } \vert ^{\alpha _{1} (x)} \biggr)\,dx} \\ &\quad \quad {} - \int _{\varOmega }{ \biggl(\frac{{C_{1} }}{{v(x)}}} \vert {u_{n} } \vert + \frac{1}{{v(x)}} \vert {u_{n} } \vert ^{\alpha _{1} (x)} \biggr)\,dx - C \\ &\quad \ge \frac{{l_{1} }}{{2v_{2} p_{2} }} \int _{\varOmega }{ \vert {\nabla u_{n} } \vert ^{p(x)} + \vert {u_{n} } \vert ^{p(x)} } \,dx - \frac{ {v_{1} C_{1} + C_{1} }}{{v_{1} }} \int _{\varOmega }{ \vert {u_{n} } \vert \,dx} \\ &\quad\quad {} - \frac{{\alpha _{1}^{-} + v_{1} }}{{\alpha _{1}^{-} v_{1} }} \int _{\varOmega }{ \vert {u_{n} } \vert ^{\alpha _{1} (x)} } \,dx - C, \end{aligned}$$

where \(\alpha _{1}^{+} = \sup_{x \in \varOmega } \alpha _{1} (x)\), \(\alpha _{1}^{-} = \inf_{x \in \varOmega } \alpha _{1} (x)\).

Since \(\alpha _{1} (x) \ll p(x)\), by the Young inequality, we have

$$\begin{aligned} \begin{aligned}[b] \int _{\varOmega }{ \vert {u_{n} } \vert ^{\alpha _{1} (x)} } \,dx &\le \int _{\varOmega }{\varepsilon \frac{{\alpha _{1} (x)}}{{p(x)}} \vert {u_{n} } \vert ^{p(x)} } + \frac{{p(x) - \alpha _{1} (x)}}{{p(x)}} \varepsilon ^{\frac{{\alpha _{1} (x)}}{{\alpha _{1} (x) - p(x)}}} \,dx \\ &\le \frac{{\alpha _{1}^{+} }}{{p_{1} }}\varepsilon \int _{\varOmega } { \vert {u_{n} } \vert ^{p(x)} } \,dx + C(\varepsilon ). \end{aligned} \end{aligned}$$
(4.4)

In (3.7) and (4.4), we choose

$$ \varepsilon _{1} = \min \biggl\{ {1\frac{{l_{1} v_{1} p_{1} }}{{4v_{2} p_{2} (v_{1} C_{1} + C_{1} )}}} \biggr\} , \quad\quad \varepsilon = \min \biggl\{ {1\frac{{l_{1} p_{1} v_{1} \alpha _{1}^{-} }}{{4v_{2} p_{2} \alpha _{1} ^{+} (\alpha _{1}^{-} + v_{1} )}}} \biggr\} . $$

Then

$$ I(u_{n} ) - \biggl\langle {I'(u_{n} ), \frac{{u_{n} }}{{v(x)}}} \biggr\rangle \ge \frac{{l_{1} }}{{4v_{2} p_{2} }} \int _{\varOmega }{ \vert {\nabla u _{n} } \vert ^{p(x)} + \vert {u_{n} } \vert ^{p(x)} } \,dx - C. $$

Similarly to (3) of Theorem 3.1, we find that \(\{u_{n}\}\) is bounded. By Lemma 3.5, we find that the functional I satisfies the (PS) condition.

From the fountain theorem, the proof of Theorem 4.1 follows immediately from (1), (2) and (3). □

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Acknowledgements

The authors are very grateful to the referees for carefully reading of the paper and for their useful comments and suggestions, which have improved the paper.

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Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11771107 and No. 11801120), and the Research Funding of Heilongjiang Province (Grant No. RCCX201716).

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Correspondence to Yingying Shan.

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Keywords

  • Variable exponent Sobolev space
  • Weak solution
  • Mountain pass theorem
  • Fountain Theorem
  • Variational method