On a p(x)-Biharmonic Kirchhoff Problem with Logarithmic Nonlinearity

Abstract

This paper is devoted to the study of a class of the $p\left(x\right)$-biharmonic Kirchhoff problem with logarithmic nonlinearity. With the help of the mountain pass theorem, the existence of a nontrivial weak solution to this problem is obtained.

1. Introduction

In this paper, we consider the following fourth-order nonlinear elliptic problem with a $p\left(x\right)$-biharmonic operator:

$$\left\{\begin{array}{cc}M\left({\delta }_{p(·)}\left(u\right)\right)\left({\Delta }_{p\left(x\right)}^{2}u+a\left(x\right){|u|}^{p\left(x\right)-2}u\right)={\lambda c\left(x\right)|u|}^{q\left(x\right)-2}u+{d\left(x\right)|u|}^{\theta p\left(x\right)-2}uln|u|\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \Delta u=u=0\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }\subset {\mathbb{R}}^N(N\ge 3)$ is a smooth and bounded domain, $\lambda$ is a positive parameter, $\theta$ is a positive real number, $p\left(x\right)$, $q\left(x\right)\in C\left(\overline{\mathsf{\Omega }}\right)$ such that $2<p\left(x\right)<\theta p\left(x\right)<q\left(x\right)<p^{*}\left(x\right)$ $\left(p^{*}\left(x\right)={\displaystyle \frac{Np\left(x\right)}{N-2p\left(x\right)}}\mathrm{if}N>2p\left(x\right),p^{*}\left(x\right)=\infty \mathrm{if}N\le 2p\left(x\right)\right)$ for any $x\in \mathsf{\Omega }$, $a\left(x\right)\in L^{\infty }\left(\mathsf{\Omega }\right)$ and $\underset{x\in \mathsf{\Omega }}{essin f} a\left(x\right)>0$, $c\left(x\right),d\left(x\right)\in C\left(\overline{\mathsf{\Omega }}\right)$ are two positive functions. ${\Delta }_{p\left(x\right)}^{2}u={\Delta (|\Delta u|}^{p\left(x\right)-2}\Delta u)$, which is a fourth-order operator called the $p\left(x\right)$-biharmonic operator. $M\left(t\right)$ is a continuous function, ${\delta }_{p(·)}\left(u\right)={\int }_{\mathsf{\Omega }}{\displaystyle \frac{1}{p\left(x\right)}}{(|\Delta u|}^{p\left(x\right)}+{a\left(x\right)|u|}^{p\left(x\right)})dx$.

The study of differential equations and variational problems with nonstandard $p\left(x\right)$-growth conditions has been a new and interesting topic. The reason for this interest comes from the fact that they have many applications in electrorheological fluids [1,2], image restoration [3], mathematical biology [4], dielectric breakdown, electrical resistivity, and polycrystal plasticity [5,6].

In recent years, interest in nonlinear Kirchhoff type problems has grown, thanks in particular to their intriguing analytical structure due to the presence of the nonlocal Kirchhoff function M, which no longer makes the equation a pointwise identity. This phenomenon poses some mathematical difficulties that are particularly fascinating to study. In fact, Problem (1) is related to the stationary problem of a model introduced by Kirchhoff [7]. To be more precise, Kirchhoff established a model given by the following equation:

$$\rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\left({\displaystyle \frac{{\rho }_0}{h}}+{\displaystyle \frac{E}{2L}}{\int }_0^L|{\displaystyle \frac{\partial u}{\partial x}}|dx\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0,$$

(2)

where $\rho ,{\rho }_0,h,E,L$ are constants, which extends the classical D’Alambert’s wave equation by considering the effects of the changes in the length of the strings during the vibrations. In two dimensions, Kirchhoff equations model the oscillations of thin plates. A typical prototype of M is given by $M\left(t\right)=a+b{t}^{\alpha -1},\mathrm{for}\mathrm{all}t\ge 0,\mathrm{where}a\ge 0,b>0\mathrm{and}\alpha >1$. When $M\left(t\right)>0$ for all $t\ge 0$, Kirchhoff problems are said to be nondegenerate. Otherwise, if $M\left(0\right)=0$ and $M\left(t\right)>0$ for all $t>0$, the Kirchhoff problems are called degenerate. The degenerate case is very appealing, and it is covered in famous well-known papers in Kirchhoff theory, including [8,9,10]. From a physical point of view, the fact that $M\left(0\right)=0$ means that the base tension of the string is zero, providing a very realistic model.

Fourth-order equations have various applications in many domains like microelectromechanical systems, surface diffusion on solids, thin film theory, and interface dynamics; we refer to [11,12,13,14,15,16,17,18,19,20]. In particular, the author in [20] considers the following non-local fourth-order weighted elliptic equation:

$$\left\{\begin{array}{cc}g\left({\int }_B(v_{\beta }{(x)|▵u|}^{{\displaystyle \frac{N}{2}}})dx\right)▵(v_{\beta }{(x)|▵u|}^{{\displaystyle \frac{N}{2}}-2}{▵u)dx=|u|}^{q-2}u+f(x,u)\hfill & \mathrm{in}B,\hfill \\ u={\displaystyle \frac{\partial u}{\partial n}}=0\hfill & \mathrm{on}\partial B,\hfill \end{array}\right.$$

(3)

where $B=B(0,1)$ is the unit open ball in $R^N$, and $q>N$, $f(x,t)$ is continuous in $B\times R$ and behaves like $exp\{\alpha t^{{\displaystyle \frac{N}{(N-2)(1-\beta )}}}\}$ as $t\to +\infty$ for some $\alpha >0$ uniformly with respect to $x\in B$. The weight $v_{\beta }\left(x\right)$ is given by

$$v_{\beta }\left(x\right)={\left(log{\displaystyle \frac{e}{|x|}}\right)}^{\beta ({\displaystyle \frac{N}{2}}-1)},\beta \in (0,1).$$

The Kirchhoff function g is positive, continuous, and verifies some mild conditions. Using the Nehari manifold method, the quantitative deformation lemma, and results from degree theory, the author established the existence of a ground-state solution. The interplay between the fourth-order equation and the variable exponent equation goes to the $p\left(x\right)$-biharmonic problems. The $p\left(x\right)$-biharmonic operator possesses more complicated nonlinearity than the p-biharmonic operator due to the fact that ${\Delta }_{p\left(x\right)}^2$ is nonhomogeneous. This fact implies some difficulties; for example, we can not use the Lagrange Multiplier theorem in many problems involving this operator.

On the other hand, logarithmic nonlinearity is widely used in partial differential equations describing mathematical and physical phenomena. Elliptic equations with logarithmic nonlinearity have also been extensively studied; see [21,22,23,24,25,26,27]. In particular, we point out that Tian in [23] investigated the existence of two local least energy solutions for fractional p-Kirchhoff problems involving logarithmic nonlinearity using the Nehari manifold approach. Until now, there have been few papers handling equations involving logarithmic nonlinearity with variable exponents. The difficulty here is the lack of logarithmic Sobolev inequality; it seems there is no logarithmic Sobolev inequality concerning the $p\left(x\right)$-biharmonic yet.

Motivated by the aforementioned cited works, there is no result for Kirchhoff-type equations combined with variable exponents, a $p\left(x\right)$-biharmonic operator, and logarithmic nonlinearity. Therefore, we will consider the existence of solutions to such equations, which differ from the work in the previously mentioned articles. We use some technical means to deal with the logarithmic nonlinearity, and with the help of the mountain pass theorem, the existence of a nontrivial weak solution for the above problem is obtained. Furthermore, we speculate that the numerical solution to this problem can be analyzed using standard methods. One of the main difficulties and innovations of the present article is that we consider the $p\left(x\right)$-biharmonic operator and logarithmic nonlinearity with variable exponents; another difficulty is that we consider the degenerate Kirchhoff equation.

To study our main result, we need to make further assumptions:

$\left(M_1\right)$$2<p^{-}\le p\left(x\right)\le p^{+}<\theta p^{-}\le \theta p\left(x\right)\le \theta p^{+}<q^{-}\le q\left(x\right)\le q^{+}<p^{*}\left(x\right)$, where $s^{-}=\underset{\overline{\mathsf{\Omega }}}{min}s\left(x\right)$, $s^{+}=\underset{\overline{\mathsf{\Omega }}}{max}s\left(x\right)$.

$\left(M_2\right)$ There are tow constants, $h_1$, $h_2$, such that

$$h_1{t}^{\theta -1}\le M\left(t\right)\le h_2{t}^{\theta -1}\mathrm{for}\mathrm{all}t\in {\mathbb{R}}^{+}.$$

We are ready to state the main result of this paper.

Theorem 1. 

Assume that the assumptions $\left(M_1\right)$ and $\left(M_2\right)$ hold. Then, there exists ${\lambda }_{*}>0$ such that for any $\lambda \in (0,{\lambda }_{*})$, Problem (1) has a nontrivial weak solution.

The rest of this paper is organized as follows. Section 2 contains some preliminary lemmas on the generalized Lebesgue–Sobolev spaces. In Section 3, Theorem 1 is proved by proving some lemmas.

2. Preliminaries

In order to investigate Problem (1), we first recall some results on the corresponding spaces. Let $\mathsf{\Omega }$ be a bounded domain of ${\mathbb{R}}^N$, $C_{+}\left(\mathsf{\Omega }\right)=\{h:h\in C\left(\overline{\mathsf{\Omega }}\right)\mathrm{and}h\left(x\right)>1,\forall x\in \overline{\mathsf{\Omega }}\}$.

For $p\left(x\right)\in C_{+}\left(\mathsf{\Omega }\right)$, we define the variable Lebesgue space $L^{p\left(x\right)}\left(\mathsf{\Omega }\right)$ as follows:

$$L^{p\left(x\right)}\left(\mathsf{\Omega }\right):=\left\{u:\mathsf{\Omega }\to \mathbb{R}\mathrm{measurable},{\int }_{\mathsf{\Omega }}{|u\left(x\right)|}^{p\left(x\right)}dx<\infty \right\}.$$

The space endowed with the Luxemburg norm

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

is a separable reflexive Banach space (see [28]).

We define the variable exponent Sobolev space $W^{m,p\left(x\right)}\left(\mathsf{\Omega }\right)$ as follows:

$$W^{m,p\left(x\right)}\left(\mathsf{\Omega }\right):=\left\{u\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right)|D^{\alpha }u\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right),|\alpha |\le m\right\},$$

where m is a positive integer and

$$D^{\alpha }u={\displaystyle \frac{{\partial }^{|\alpha |}}{\partial x_1^{{\alpha }_1}\dots \partial x_N^{{\alpha }_N}}}u,$$

where $\alpha =({\alpha }_1,\dots ,{\alpha }_N)$ is a multi-index and $|\alpha |={\sum }_{i=1}^N{\alpha }_i$. The space $W^{m,p\left(x\right)}\left(\mathsf{\Omega }\right)$, equipped with the norm

$${\parallel u\parallel }_{m,p\left(x\right)}=\sum _{|\alpha |\le m}{|D^{\alpha }u|}_{p\left(x\right)},$$

becomes a separable, reflexive, and uniformly convex Banach space (see [28]). We denote by $W_0^{m,p\left(x\right)}\left(\mathsf{\Omega }\right)$ the closure of $C_0^{\infty }\left(\mathsf{\Omega }\right)$ in $W^{m,p\left(x\right)}\left(\mathsf{\Omega }\right)$.

Proposition 1 

([28]). The conjugate space of $L^{p\left(x\right)}\left(\mathsf{\Omega }\right)$ is $L^{q\left(x\right)}\left(\mathsf{\Omega }\right)$ with ${\displaystyle \frac{1}{p\left(x\right)}}+{\displaystyle \frac{1}{q\left(x\right)}}=1$. For $u\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right)$ and $v\in L^{q\left(x\right)}\left(\mathsf{\Omega }\right)$, we have

$$|{\int }_{\mathsf{\Omega }}u\left(x\right)v\left(x\right)dx|\le \left({\displaystyle \frac{1}{p^{-}}}+{\displaystyle \frac{1}{q^{-}}}\right){|u|}_{p\left(x\right)}{|v|}_{q\left(x\right)}\le {2|u|}_{p\left(x\right)}{|v|}_{q\left(x\right)}.$$

Proposition 2 

([28]). Let ${\rho }_{p\left(x\right)}\left(u\right)={\int }_{\mathsf{\Omega }}{|u|}^{p\left(x\right)}dx$. For all $u,u_n\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right)$, we have

(1) ${|u|}_{p\left(x\right)}<1\left(resp.=1,>1\right)⟺{\rho }_{p\left(x\right)}\left(u\right)<1\left(resp.=1,>1\right)$.

(2) $min\left({|u|}_{p\left(x\right)}^{p^{-}},{|u|}_{p\left(x\right)}^{p^{+}}\right)\le {\rho }_{p\left(x\right)}\left(u\right)\le max\left({|u|}_{p\left(x\right)}^{p^{-}},{|u|}_{p\left(x\right)}^{p^{+}}\right)$.

(3) $\underset{n\to \infty }{lim}{\rho }_{p\left(x\right)}(u_n-u)=0⟺\underset{n\to \infty }{lim}{|u_n-u|}_{p\left(x\right)}=0$.

Proposition 3 

([29]). Let p(x) and q(x) be two measurable functions such that $p\left(x\right)\in L^{\infty }\left(\mathsf{\Omega }\right)$, and $1\le p\left(x\right)q\left(x\right)<\infty$, for a.e. $x\in \mathsf{\Omega }$. Let $u\in L^{q\left(x\right)}\left(\mathsf{\Omega }\right)$, $u\ne 0$. Then,

$$min\left({|u|}_{p\left(x\right)q\left(x\right)}^{p^{+}},{|u|}_{p\left(x\right)q\left(x\right)}^{p^{-}}\right)\le |{|u|}^{p\left(x\right)}{|}_{q\left(x\right)}\le max\left({|u|}_{p\left(x\right)q\left(x\right)}^{p^{+}},{|u|}_{p\left(x\right)q\left(x\right)}^{p^{-}}\right).$$

For more details concerning the modular, one can see [30].

Throughout this paper, we let $X:=W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)\cap W^{2,p\left(x\right)}\left(\mathsf{\Omega }\right)$. For $u\in X$, we define

$${\parallel u\parallel }_X={\parallel u\parallel }_{1,p\left(x\right)}+{\parallel u\parallel }_{2,p\left(x\right)}.$$

Moreover, it is well known that if $1<p^{-}\le p^{+}<\infty$, the space $(X,\parallel ·\parallel )$ is a separable and reflexive Banach space, and ${\parallel u\parallel }_X$ and ${|\Delta u|}_{p\left(x\right)}$ are two equivalent norms on X (see [28,31]).

Let

$$\parallel u\parallel =inf\left\{\tau >0:{\int }_{\mathsf{\Omega }}\left(|{\displaystyle \frac{\Delta u}{\tau }}{|}^{p\left(x\right)}+a\left(x\right)|{\displaystyle \frac{u}{\tau }}{|}^{p\left(x\right)}\right)dx\le 1\right\}.$$

Since $a\left(x\right)\in L^{\infty }\left(\mathsf{\Omega }\right)$ and $ess \underset{x\in \mathsf{\Omega }}{inf } a\left(x\right)>0$, we deduce that $\parallel u\parallel$ is equivalent to the norms ${\parallel u\parallel }_X$ and ${|\Delta u|}_{p\left(x\right)}$ in X. In our paper, we will use the norm $\parallel ·\parallel$. The modular is defined as ${\rho }_a\left(u\right):X\to \mathbb{R}$ by

$${\rho }_a\left(u\right)={\int }_{\mathsf{\Omega }}\left({|\Delta u|}^{p\left(x\right)}+a\left(x\right){|u|}^{p\left(x\right)}\right)dx,$$

which satisfies the same properties as Proposition 2. Accordingly, we have the following property.

Proposition 4. 

For all $u,u_n\in X$, we have

(1) $\parallel u\parallel <1(resp.=1,>1)⟺{\rho }_a\left(u\right)<1(resp.=1,>1)$.

(2) $min\left({\parallel u\parallel }^{p^{-}},{\parallel u\parallel }^{p^{+}}\right)\le {\rho }_a\left(u\right)\le max\left({\parallel u\parallel }^{p^{-}},{\parallel u\parallel }^{p^{+}}\right)$.

(3) $\parallel u_n\parallel \to 0(respectively,\to \infty )⟺{\rho }_a\left(u_n\right)\to 0(respectively,\to \infty )$(as $n\to \infty$).

Proposition 5 

([32]). The modular functions have the following properties:

(1) ${\rho }_a:X\to \mathbb{R}$ is sequentially weakly lower semi-continuous, ${\rho }_a\in C^1(X,\mathbb{R})$.

(2) The mapping ${\rho }_a^{\prime }:X\to X^{*}$ is a strictly monotone, bounded homeomorphism and is of type $\left(S_{+}\right)$; that is, if $u_n\rightharpoonup u$ and $\underset{n\to +\infty }{limsup}{\rho }_a^{\prime }\left(u_n\right)(u_n-u)\le 0$, then $u_n\to u$.

Proposition 6 

([33]). Let $p\left(x\right),q\left(x\right)\in C_{+}\left(\mathsf{\Omega }\right)$ in which $p\left(x\right)\le q\left(x\right)\le p^{*}\left(x\right)$ a.e. $x\in \overline{\mathsf{\Omega }}$. Then, this is a continuous embedding $X\hookrightarrow L^{q\left(x\right)}\left(\mathsf{\Omega }\right)$. This embedding is compact if $q\left(x\right)<p^{*}\left(x\right)$.

3. Proof of the Result

In this section, we give the proof of Theorem 1. For this, we first need some technical lemmas to handle the logarithmic nonlinearity. According to [34], there are the following lemmas.

Lemma 1 

([34]). For any $\sigma >0$, we have

(i) $t^{\sigma }|lnt|\le {\displaystyle \frac{1}{e\sigma }}$ for all $t\in (0,1]$;

(ii) $lnt\le {\displaystyle \frac{t^{\sigma }}{e\sigma }}$ for all $t>1$.

Lemma 2. 

Let $u\in X\setminus \left\{0\right\}.$ Then,

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln|u|dx\le Cmax\left\{{\parallel u\parallel }^{\theta p^{-}},{\parallel u\parallel }^{\theta p^{+}}\right\}+ln(\parallel u\parallel ){\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}dx,$$

where $C>0$ is a suitable constant.

Proof. 

To prove our results, we only need to prove the following inequalities:

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln{\displaystyle \frac{|u|}{\parallel u\parallel }}dx\le Cmax\left\{{\parallel u\parallel }^{\theta p^{-}},{\parallel u\parallel }^{\theta p^{+}}\right\}.$$

We set ${\mathsf{\Omega }}_1=\{x\in \mathsf{\Omega }:|u\left(x\right)|\le \parallel u\parallel \}$ and ${\mathsf{\Omega }}_2=\{x\in \mathsf{\Omega }:|u\left(x\right)|>\parallel u\parallel \}$. Then,

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln{\displaystyle \frac{|u|}{\parallel u\parallel }}dx={\int }_{{\mathsf{\Omega }}_1}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln{\displaystyle \frac{|u|}{\parallel u\parallel }}dx+{\int }_{{\mathsf{\Omega }}_2}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln{\displaystyle \frac{|u|}{\parallel u\parallel }}dx.$$

Hence, using Lemma 1, we obtain the following:

$$\begin{array}{cc}\hfill {\int }_{{\mathsf{\Omega }}_1}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln{\displaystyle \frac{|u|}{\parallel u\parallel }}dx& \le {\displaystyle \frac{d^{+}}{\theta p^{-}}}max\left\{{\parallel u\parallel }^{\theta p^{-}},{\parallel u\parallel }^{\theta p^{+}}\right\}{\int }_{{\mathsf{\Omega }}_1}{\left({\displaystyle \frac{|u|}{\parallel u\parallel }}\right)}^{\theta p\left(x\right)}|ln{\displaystyle \frac{|u|}{\parallel u\parallel }}|dx\hfill \\ \hfill & \le {\displaystyle \frac{d^{+}}{\theta p^{-}}}max\left\{{\parallel u\parallel }^{\theta p^{-}},{\parallel u\parallel }^{\theta p^{+}}\right\}{\int }_{{\mathsf{\Omega }}_1}{\displaystyle \frac{1}{e\theta p\left(x\right)}}dx\hfill \\ \hfill & \le {\displaystyle \frac{|\mathsf{\Omega }|d^{+}}{e{\left(\theta p^{-}\right)}^2}}max\left\{{\parallel u\parallel }^{\theta p^{-}},{\parallel u\parallel }^{\theta p^{+}}\right\}.\hfill \end{array}$$

(4)

Next, we choose $ϵ>0$ sufficiently small such that $\sigma ={\left(p^{*}\right)}^{-}-ϵ-\theta p^{+}>0$. Using Lemma 1 (ii) and the Sobolev embedding inequality, we obtain the following:

$$\begin{array}{cc}\hfill {\int }_{{\mathsf{\Omega }}_2}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln{\displaystyle \frac{|u|}{\parallel u\parallel }}dx& \le {\displaystyle \frac{d^{+}}{\theta p^{-}}}{\int }_{{\mathsf{\Omega }}_2}{|u|}^{\theta p\left(x\right)}ln{\displaystyle \frac{|u|}{\parallel u\parallel }}dx\hfill \\ \hfill & \le {\displaystyle \frac{d^{+}}{e({\left(p^{*}\right)}^{-}-ϵ-\theta p^{+})\theta p^{-}}}{\int }_{{\mathsf{\Omega }}_2}{|u|}^{\theta p\left(x\right)}{\left({\displaystyle \frac{|u|}{\parallel u\parallel }}\right)}^{{\left(p^{*}\right)}^{-}-ϵ-\theta p^{+}}dx\hfill \\ \hfill & \le {\displaystyle \frac{C_1{d}^{+}}{e({\left(p^{*}\right)}^{-}-ϵ-\theta p^{+})\theta p^{-}}}{max\{\parallel u\parallel }^{\theta p^{-}}{,\parallel u\parallel }^{\theta p^{+}}\},\hfill \end{array}$$

(5)

where $C_1>0$. Combining (4) and (5), we get the result. This ends the proof. □

A function $u\in X$ is a weak solution to Problem (1) if

$$\begin{array}{cc}\hfill M\left({\delta }_{p(·)}\left(u\right)\right){\int }_{\mathsf{\Omega }}({|\Delta u|}^{p\left(x\right)-2}\Delta u\Delta v& +{a\left(x\right)|u|}^{p\left(x\right)-2}uv)dx-\lambda {\int }_{\mathsf{\Omega }}c\left(x\right){|u|}^{q\left(x\right)-2}uvdx\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)-2}uln|u|vdx=0,\hfill \end{array}$$

for any $v\in X$.

The corresponding energy functional of Problem (1) is defined as follows:

$$J_{\lambda }\left(u\right)=\widehat{M}\left({\delta }_{p(·)}\left(u\right)\right)-\lambda {\int }_{\mathsf{\Omega }}{\displaystyle \frac{c\left(x\right)}{q\left(x\right)}}{|u|}^{q\left(x\right)}dx-{\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln|u|dx+{\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{{\left(\theta p\left(x\right)\right)}^2}}{|u|}^{\theta p\left(x\right)}dx,$$

where $\widehat{M}\left(t\right)={\int }_0^{t}M\left(\tau \right)d\tau$.

Lemma 3. 

Assume that Conditions $\left(M_1\right)$-$\left(M_2\right)$ hold, and let $J_{\lambda }\left(0\right)=0$. Then, the functional $J_{\lambda }$ is well defined in X. Moreover, $J_{\lambda }\in C^1(X,\mathbb{R})$ with the derivative given as follows:

$$\begin{array}{cc}\hfill \langle J_{\lambda }^{\prime }\left(u\right),v\rangle =M\left({\delta }_{p(·)}\left(u\right)\right)& {\int }_{\mathsf{\Omega }}\left({|\Delta u|}^{p\left(x\right)-2}\Delta u\Delta v+a\left(x\right){|u|}^{p\left(x\right)-2}uv\right)dx\hfill \\ \hfill & -\lambda {\int }_{\mathsf{\Omega }}{c\left(x\right)|u|}^{q\left(x\right)-2}uvdx-{\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)-2}uln|u|vdx\hfill \end{array}$$

for any $v\in X$.

Proof. 

Let $u\in X$; based on $\left(M_2\right)$, we have the following:

$$\begin{array}{cc}\hfill J_{\lambda }\left(u\right)\le {\displaystyle \frac{h_2}{\theta {\left(p^{+}\right)}^{\theta }}}{\left({\rho }_a\left(u\right)\right)}^{\theta }& -{\displaystyle \frac{\lambda }{q^{+}}}{\int }_{\mathsf{\Omega }}c\left(x\right){|u|}^{q\left(x\right)}dx\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln|u|dx+{\displaystyle \frac{1}{{\left(\theta p^{-}\right)}^2}}{\int }_{\mathsf{\Omega }}d\left(x\right){|u|}^{\theta p\left(x\right)}dx.\hfill \end{array}$$

On the one hand, using Proposition 4, and based on the definition of norm $\parallel ·\parallel$, we obtain the following:

$${\rho }_a\left(u\right)\le max\left\{{\parallel u\parallel }^{p^{-}},{\parallel u\parallel }^{p^{+}}\right\}<+\infty .$$

On the other hand, using Proposition 6, we have the following:

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}c\left(x\right){|u|}^{q\left(x\right)}dx\le c^{+}max\left\{{|u|}_{q\left(x\right)}^{q^{-}},{|u|}_{q\left(x\right)}^{q^{+}}\right\}\le c^{+}max\left\{{\parallel u\parallel }^{q^{-}},{\parallel u\parallel }^{q^{+}}\right\}<+\infty .\end{array}$$

Similarly, we can prove that

$${\int }_{\mathsf{\Omega }}d\left(x\right){|u|}^{\theta p\left(x\right)}dx<+\infty .$$

Furthermore, from Lemma 2 we have the following:

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln|u|dx\le Cmax\left\{{\parallel u\parallel }^{\theta p^{-}},{\parallel u\parallel }^{\theta p^{+}}\right\}+ln\left(\parallel u\parallel \right){\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}dx<+\infty .$$

Hence, the above four inequalities imply that $J_{\lambda }$ is well defined in X. It remains to be proved that the functional $J_{\lambda }$ is of class $C^1(X,\mathbb{R})$. For this aim, for every $u\in X$, we define the following:

$$\begin{array}{cc}\hfill & G\left(u\right)=\widehat{M}\left({\delta }_{p(·)}\left(u\right)\right),\hfill \\ \hfill & H\left(u\right)={\int }_{\mathsf{\Omega }}{\displaystyle \frac{c\left(x\right)}{q\left(x\right)}}{|u|}^{q\left(x\right)}dx+{\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}ln|u|dx-{\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{{\left(\theta p\left(x\right)\right)}^2}}{|u|}^{\theta p\left(x\right)}dx.\hfill \end{array}$$

Firstly, using some simple computations, we can show that $G\in C^1(X,\mathbb{R})$ and

$$\langle G^{\prime }\left(u\right),v\rangle =M\left({\delta }_{p(·)}\left(u\right)\right){\int }_{\mathsf{\Omega }}\left({|\Delta u|}^{p\left(x\right)-2}\Delta u\Delta v+a\left(x\right){|u|}^{p\left(x\right)-2}uv\right)dx.$$

Second, it is easy to see that H is Gateaux-differentiable in X and

$$\langle H^{\prime }\left(u\right),v\rangle ={\int }_{\mathsf{\Omega }}{c\left(x\right)|u|}^{q\left(x\right)-2}uvdx+{\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)-2}uln|u|dx$$

for all $u,v\in X$. Next, we want to show the continuity of the Gateaux derivative of H. Now, let $\left\{u_n\right\}\in X$ and $u\in X$ such that $u_n\to u$ in X as $n\to +\infty$. For any $v\in X$ and $\parallel v\parallel \le 1$, based on the Hölder inequality, we have the following:

$$\begin{array}{cc}\hfill \left|\langle H^{\prime }\left(u_n\right)-H^{\prime }\left(u\right),v\rangle \right|& \le {\int }_{\mathsf{\Omega }}c\left(x\right)\left||u_n{|}^{q\left(x\right)-2}{u}_n-{|u|}^{q\left(x\right)-2}u\right||v|dx\hfill \\ \hfill & +{\int }_{\mathsf{\Omega }}d\left(x\right)\left||u_n{|}^{\theta p\left(x\right)-2}{u}_{n}ln|u_n{|-|u|}^{\theta p\left(x\right)-2}uln|u|\right||v|dx\hfill \\ \hfill & =A_1+A_2.\hfill \end{array}$$

Based on Theorem A.2 of [35], we obtain the following as $n\to \infty$:

$$|u_n{|}^{q\left(x\right)-2}{u}_n\to {|u|}^{q\left(x\right)-2}u\mathrm{in}{L}^{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}\left(\mathsf{\Omega }\right).$$

Hence, we obtain the following:

$$\begin{array}{c}\hfill A_1\le c^{+}{∥|u_n{|}^{q\left(x\right)-2}{u}_n-{|u|}^{q\left(x\right)-2}u∥}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}{\parallel v\parallel }_{q\left(x\right)}\to 0.\end{array}$$

(6)

Let $\sigma \in \left(0,{\left(p^{*}\right)}^{-}-\theta p^{+}\right)$. For any measurable subset ${\mathsf{\Omega }}^{\prime }\subset \mathsf{\Omega }$, based on Lemma 1 we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathsf{\Omega }}^{\prime }}d\left(x\right)|u_n{|}^{\theta p\left(x\right)}ln|u_n|dx\hfill \\ \hfill & ={\int }_{{\mathsf{\Omega }}^{\prime }\cap \{|u_n|\le 1\}}d\left(x\right)|u_n{|}^{\theta p\left(x\right)}ln|u_n|dx+{\int }_{{\mathsf{\Omega }}^{\prime }\cap \{|u_n|>1\}}d\left(x\right)|u_n{|}^{\theta p\left(x\right)}ln|u_n|dx\hfill \\ \hfill & \le {\displaystyle \frac{|{\mathsf{\Omega }}^{\prime }|d^{+}}{e\theta p^{-}}}+{\displaystyle \frac{d^{+}}{e\sigma }}{\int }_{{\mathsf{\Omega }}^{\prime }}{|u_n|}^{\theta p^{+}+\sigma }dx\hfill \\ \hfill & \le {\displaystyle \frac{|{\mathsf{\Omega }}^{\prime }|d^{+}}{e\theta p^{-}}}+M{\displaystyle \frac{|{\mathsf{\Omega }}^{\prime }|C{d}^{+}}{e\sigma }},\hfill \end{array}$$

with $M=sup|u_n{|}^{\theta p^{+}+\sigma }<\infty$, which implies that the sequence $\{d\left(x\right)|u_n{|}^{\theta p\left(x\right)}ln|u_n{|\}}_{n\ge 1}$ is uniformly bounded and equi-integrable in $L^1\left(\mathsf{\Omega }\right)$. Moreover, it is easy to see that as $n\to \infty$, we have the following:

$$d\left(x\right)|u_n{|}^{\theta p\left(x\right)}ln|u_n{|\to d\left(x\right)|u|}^{\theta p\left(x\right)}ln|u|a.e.\mathrm{in}\mathsf{\Omega }.$$

Thus, Vitali’s convergence theorem yields that

$$\underset{n\to \infty }{lim}{\int }_{\mathsf{\Omega }}d\left(x\right)|u_n{|}^{\theta p\left(x\right)}ln|u_n|dx={\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)}ln|u|dx.$$

Similarly, we can prove that

$$\underset{n\to \infty }{lim}{\int }_{\mathsf{\Omega }}d\left(x\right)|u_n{|}^{\theta p\left(x\right)-2}{u}_{n}ln|u_n|dx={\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)-2}uln|u|dx.$$

Hence, we obtain the following as $n\to \infty$:

$$A_2={\int }_{\mathsf{\Omega }}d\left(x\right)\left||u_n{|}^{\theta p\left(x\right)-2}{u}_{n}ln|u_n{|-|u|}^{\theta p\left(x\right)-2}uln|u|\right||v|dx\to 0.$$

(7)

Consequently, from (6) and (7), we conclude that

$$\underset{n\to \infty }{lim}\parallel H^{\prime }\left(u_n\right)-H^{\prime }\left(u\right)\parallel =0.$$

Hence, $H\in C^1(X,\mathbb{R})$. This ends the proof of Lemma 3. □

Thus, according to Lemma 3 given above, we determine that the critical points of $J_{\lambda }$ are weak solutions of (1).

Lemma 4. 

Assume that the conditions $\left(M_1\right)$ and $\left(M_2\right)$ hold. Then, there exist ${\lambda }_{*},\rho ,\alpha >0$ such that for $\lambda \in (0,{\lambda }_{*})$, we have $J_{\lambda }\left(u\right)\ge \alpha$ for all $u\in X$ with $\parallel u\parallel =\rho$.

Proof. 

For any $u\in X$ with $\parallel u\parallel =\rho \in (0,1)$, from $\left(M_2\right)$ and Lemma 2, we have the following:

$$\begin{array}{cc}\hfill J_{\lambda }\left(u\right)& \ge {\displaystyle \frac{h_1}{\theta {\left(p^{+}\right)}^{\theta }}}{\left({\rho }_a\left(u\right)\right)}^{\theta }-{\displaystyle \frac{\lambda }{q^{-}}}{\int }_{\mathsf{\Omega }}{c\left(x\right)|u|}^{q\left(x\right)}dx-{Cmax\{\parallel u\parallel }^{\theta p^{-}}{,\parallel u\parallel }^{\theta p^{+}}\}\hfill \\ \hfill & -ln(\parallel u\parallel ){\int }_{\mathsf{\Omega }}{\displaystyle \frac{d\left(x\right)}{\theta p\left(x\right)}}{|u|}^{\theta p\left(x\right)}dx\hfill \\ \hfill & \ge {\displaystyle \frac{h_1}{\theta {\left(p^{+}\right)}^{\theta }}}{\left({\rho }_a\left(u\right)\right)}^{\theta }-{\displaystyle \frac{\lambda c^{+}}{q^{-}}}max\left\{{|u|}_{q\left(x\right)}^{q^{-}},{|u|}_{q\left(x\right)}^{q^{+}}\right\}-C{\parallel u\parallel }^{\theta p^{-}}\hfill \\ \hfill & \ge {\displaystyle \frac{h_1}{\theta {\left(p^{+}\right)}^{\theta }}}{\parallel u\parallel }^{\theta p^{+}}-{\displaystyle \frac{\lambda c^{+}}{q^{-}}}{\parallel u\parallel }^{q^{-}}-C{\parallel u\parallel }^{\theta p^{-}}\hfill \\ \hfill & \ge \left({\displaystyle \frac{h_1}{\theta {\left(p^{+}\right)}^{\theta }}}-C{\rho }^{\theta p^{-}-\theta p^{+}}\right){\rho }^{\theta p^{+}}-{\displaystyle \frac{\lambda c^{+}}{q^{-}}}{\rho }^{q^{-}}.\hfill \end{array}$$

We choose ${\rho }_0\in (0,1)$ such that ${\displaystyle \frac{h_1}{\theta {\left(p^{+}\right)}^{\theta }}}-C{\rho }_0^{\theta p^{-}-\theta p^{+}}\ge {\displaystyle \frac{h_1}{2\theta {\left(p^{+}\right)}^{\theta }}}$, and we obtain the following:

$$\begin{array}{cc}\hfill J_{\lambda }\left(u\right)& \ge {\displaystyle \frac{h_1}{2\theta {\left(p^{+}\right)}^{\theta }}}{\rho }_0^{\theta p^{+}}-{\displaystyle \frac{\lambda c^{+}}{q^{-}}}{\rho }_0^{q^{-}}\hfill \\ \hfill & \ge \left({\displaystyle \frac{h_1}{2\theta {\left(p^{+}\right)}^{\theta }}}-{\displaystyle \frac{\lambda c^{+}}{q^{-}}}\right){\rho }_0^{\theta p^{+}}.\hfill \end{array}$$

Hence, let ${\lambda }_{*}={\displaystyle \frac{h_1{q}^{-}}{2c^{+}\theta {\left(p^{+}\right)}^{\theta }}}$; then, $J_{\lambda }\left(u\right)\ge \alpha >0$ for any $\lambda \in (0,{\lambda }_{*})$. □

Lemma 5. 

Assume that Conditions $\left(M_1\right)$ and $\left(M_2\right)$ hold; then, there exist $\omega \in X$ such that $\parallel \omega \parallel >0$ and $J_{\lambda }\left(\omega \right)<0$.

Proof. 

Let $u\in X\setminus \left\{0\right\}$. Based on $\left(M_2\right)$, we have the following:

$$J_{\lambda }\left(u\right)\le {\displaystyle \frac{h_2}{\theta {\left(p^{-}\right)}^{\theta }}}{\left({\rho }_a\left(u\right)\right)}^{\theta }-{\displaystyle \frac{\lambda }{q^{+}}}{\int }_{\mathsf{\Omega }}{c\left(x\right)|u|}^{q\left(x\right)}dx+{\displaystyle \frac{1}{{\left(\theta p^{-}\right)}^2}}{\int }_{\mathsf{\Omega }}d\left(x\right){|u|}^{\theta p\left(x\right)}dx.$$

Then, fixing $u\ne 0$ and choosing $t>1$, we obtain the following:

$$\begin{array}{cc}\hfill J_{\lambda }\left(tu\right)& \le {\displaystyle \frac{t^{\theta p^{+}}{h}_2}{\theta {\left(p^{-}\right)}^{\theta }}}{\left({\rho }_a\left(u\right)\right)}^{\theta }-{\displaystyle \frac{t^{q^{-}}\lambda }{q^{+}}}{\int }_{\mathsf{\Omega }}{c\left(x\right)|u|}^{q\left(x\right)}dx+{\displaystyle \frac{t^{\theta p^{+}}}{{\left(\theta p^{-}\right)}^2}}{\int }_{\mathsf{\Omega }}d\left(x\right){|u|}^{\theta p\left(x\right)}dx\hfill \\ \hfill & =t^{\theta p^{+}}\left({\displaystyle \frac{h_2}{\theta {\left(p^{-}\right)}^{\theta }}}{\left({\rho }_a\left(u\right)\right)}^{\theta }-{\displaystyle \frac{t^{q^{-}-\theta p^{+}}\lambda }{q^{+}}}{\int }_{\mathsf{\Omega }}{c\left(x\right)|u|}^{q\left(x\right)}dx+{\displaystyle \frac{1}{{\left(\theta p^{-}\right)}^2}}{\int }_{\mathsf{\Omega }}d\left(x\right){|u|}^{\theta p\left(x\right)}dx\right).\hfill \end{array}$$

Since $q^{-}>\theta p^{+}$, we deduce that $J_{\lambda }\left(tu\right)\to -\infty$ as $t\to +\infty$. Hence, for sufficiently large $t>1$, we can let $\omega =tu$ such that $J_{\lambda }\left(\omega \right)<0$. □

Lemma 6. 

Suppose that the conditions in Theorem 1 hold. Then, the functional $J_{\lambda }$ satisfies the ${\left(PS\right)}_c$ condition.

Proof. 

Let $\left\{u_n\right\}\in X$ be a ${\left(PS\right)}_c$ sequence, i.e.,

$$J_{\lambda }\left(u_n\right)\to c\mathrm{and}{J}_{\lambda }^{\prime }\left(u_n\right)\to 0\mathrm{in}{X}^{*}.$$

(8)

We first prove that $\left\{u_n\right\}$ is bounded in X. Arguing by contradiction, if $\left\{u_n\right\}$ is unbounded in X, up to a subsequence, we may assume that $\parallel u_n\parallel \to +\infty$ as $n\to \infty$. Then, according to Conditions $\left(M_1\right)$ and $\left(M_2\right)$, for n large enough, we obtain the following:

$$\begin{array}{cc}\hfill & J_{\lambda }\left(u_n\right)-{\displaystyle \frac{1}{\theta p^{-}}}\langle J_{\lambda }^{\prime }\left(u_n\right),u_n\rangle \hfill \\ \hfill & \ge -\left({\displaystyle \frac{h_2}{\theta {\left(p^{-}\right)}^{\theta }}}-{\displaystyle \frac{h_1}{\theta {\left(p^{+}\right)}^{\theta }}}\right){\left({\rho }_a\left(u_n\right)\right)}^{\theta }+\lambda \left({\displaystyle \frac{1}{\theta p^{-}}}-{\displaystyle \frac{1}{q^{-}}}\right){\int }_{\mathsf{\Omega }}c\left(x\right){|u_n|}^{q\left(x\right)}dx\hfill \\ \hfill & +{\displaystyle \frac{1}{{\left(\theta p^{+}\right)}^2}}{\int }_{\mathsf{\Omega }}d\left(x\right){|u_n|}^{\theta p\left(x\right)}dx\hfill \\ \hfill & \ge -\left({\displaystyle \frac{h_2}{\theta {\left(p^{-}\right)}^{\theta }}}-{\displaystyle \frac{h_1}{\theta {\left(p^{+}\right)}^{\theta }}}\right)\parallel u_n{\parallel }^{\theta p^{+}}+\lambda \left({\displaystyle \frac{1}{\theta p^{-}}}-{\displaystyle \frac{1}{q^{-}}}\right)c^{-}{\parallel u_n\parallel }^{q^{-}}\hfill \\ \hfill & +{\displaystyle \frac{d^{-}}{{\left(\theta p^{+}\right)}^2}}{\parallel u_n\parallel }^{\theta p^{-}}.\hfill \end{array}$$

Since $q^{-}>\theta p^{+}$, then $c\ge +\infty$; we obtain a contradiction. Then, $\left\{u_n\right\}$ is bounded in X. Now, we prove that $\left\{u_n\right\}$ has a convergent subsequence in X. In view of Proposition 6 and the reflexivity of X, there exists a subsequence, still denoted by $\left\{u_n\right\}$, and $u\in X$ such that

$$u_n\rightharpoonup u\mathrm{in}X,u_n\to u\mathrm{in}{L}^{r\left(x\right)}\left(\mathsf{\Omega }\right)(1<r\left(x\right)<p^{*}\left(x\right)),u_n\left(x\right)\to u\left(x\right)a.e.\mathrm{in}\mathsf{\Omega }.$$

(9)

Based on (8), we obtain the following:

$$\langle J_{\lambda }^{\prime }\left(u_n\right),u_n-u\rangle \to 0.$$

Moreover,

$$\begin{array}{cc}\hfill M& \left({\delta }_{p(·)}\left(u_n\right)\right){\int }_{\mathsf{\Omega }}\left(|\Delta u_n{|}^{p\left(x\right)-2}\Delta u_n(\Delta u_n-\Delta u)+a\left(x\right){|u_n|}^{p\left(x\right)-2}{u}_n(u_n-u)\right)dx\hfill \\ \hfill & =\langle J_{\lambda }^{\prime }\left(u_n\right),u_n-u\rangle -\lambda {\int }_{\mathsf{\Omega }}c\left(x\right){|u_n|}^{q\left(x\right)-2}{u}_n(u_n-u)dx\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}d\left(x\right)|u_n{|}^{\theta p\left(x\right)-2}{u}_{n}ln|u_n|(u_n-u)dx.\hfill \end{array}$$

Using the Hölder inequality, we get

$$\begin{array}{cc}\hfill |{\int }_{\mathsf{\Omega }}{|u_n|}^{q\left(x\right)-2}{u}_n(u_n-u)dx|& \le {\int }_{\mathsf{\Omega }}|u_n{|}^{q\left(x\right)-1}|u_n-u|dx\hfill \\ \hfill & \le 2||u_n{|}^{q\left(x\right)-1}{|}_{{\displaystyle \frac{q\left(x\right)}{q\left(x\right)-1}}}{|u_n-u|}_{q\left(x\right)}\hfill \\ \hfill & \le 2max\left\{|u_n{|}_{q\left(x\right)}^{q^{+}-1},{|u_n|}_{q\left(x\right)}^{q^{-}-1}\right\}{|u_n-u|}_{q\left(x\right)},\hfill \end{array}$$

then, based on (9),

$${\int }_{\mathsf{\Omega }}c\left(x\right){|u_n|}^{q\left(x\right)-2}{u}_n(u_n-u)dx\to 0\mathrm{as}n\to +\infty .$$

(10)

In the proof of Lemma 3, we obtain the following:

$$\underset{n\to \infty }{lim}{\int }_{\mathsf{\Omega }}d\left(x\right)|u_n{|}^{\theta p\left(x\right)}ln|u_n|dx={\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)}ln|u|dx.$$

(11)

Similarly, we can prove that

$$\underset{n\to \infty }{lim}{\int }_{\mathsf{\Omega }}d\left(x\right)u|u_n{|}^{\theta p\left(x\right)-2}{u}_{n}ln|u_n|dx={\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)}ln|u|dx,$$

(12)

and

$$\underset{n\to \infty }{lim}{\int }_{\mathsf{\Omega }}d\left(x\right)u_n|u_n{|}^{\theta p\left(x\right)-2}uln|u|dx={\int }_{\mathsf{\Omega }}{d\left(x\right)|u|}^{\theta p\left(x\right)}ln|u|dx.$$

(13)

Consequently, from (11)–(13), we conclude that

$$\underset{n\to \infty }{lim}{\int }_{\mathsf{\Omega }}d\left(x\right)|u_n{|}^{\theta p\left(x\right)-2}{u}_n(u_n-u)ln|u_n|dx={\int }_{\mathsf{\Omega }}d\left(x\right){|u|}^{\theta p\left(x\right)-2}u(u_n-u)ln|u|dx.$$

(14)

From (9), (10), and (14), we get

$$M\left({\delta }_{p(·)}\left(u_n\right)\right){\int }_{\mathsf{\Omega }}\left(|\Delta u_n{|}^{p\left(x\right)-2}\Delta u_n(\Delta u_n-\Delta u)+a\left(x\right){|u_n|}^{p\left(x\right)-2}{u}_n(u_n-u)\right)dx\to 0$$

(15)

as $n\to +\infty$. As $\left\{u_n\right\}$ is bounded in X, we suppose that

$${\delta }_{p(·)}\left(u_n\right)\to \kappa \ge 0,\mathrm{as}n\to \infty .$$

If $\kappa =0$, then $\left\{u_n\right\}$ strongly converges to $u=0$ in X, and the proof is complete.

If $\kappa >0$, because the function M is continuous, we know that

$$M\left({\delta }_{p(·)}\left(u_n\right)\right)\to M\left(\kappa \right)>0,\mathrm{as}n\to \infty .$$

(16)

From (10)–(16), we obtain the following:

$$\underset{n\to \infty }{lim}{\int }_{\mathsf{\Omega }}\left(|\Delta u_n{|}^{p\left(x\right)-2}\Delta u_n(\Delta u_n-\Delta u)+a\left(x\right){|u_n|}^{p\left(x\right)-2}{u}_n(u_n-u)\right)dx=0.$$

Therefore, based on Proposition 5, we can now deduce that $u_n\to u$ in X, which means that $J_{\lambda }$ satisfies the ${\left(PS\right)}_c$ condition. This completes the proof. □

Proof of Theorem 1. 

Thanks to Lemmas 4–6, there exists ${\lambda }_{*}>0$ such that for all $\lambda \in (0,{\lambda }_{*})$, $J_{\lambda }$ satisfies the mountain pass geometry and ${\left(PS\right)}_c$ condition. By employing the mountain pass theorem in [35], we infer that there exists a critical point $u_0\in E$ of $J_{\lambda }$ with $J_{\lambda }\left(u_0\right)=\overline{c}>0=J_{\lambda }\left(0\right)$. Hence, $u_0$ is a nontrivial weak solution of Problem (1). □