Nontrivial Solutions and Asymptotic Behavior of Variable Critical p(x)-Laplace Equations with Negative Nonlocal Kirchhoff Terms

Abstract

We study a Dirichlet problem for a $p\left(x\right)$-Kirchhoff equation with negative nonlocal structure and variable critical nonlinearity. The modified Kirchhoff term and variable critical exponent break Sobolev compactness and complicate Palais–Smale sequence analysis. Using a truncation argument, the mountain pass theorem and the variable exponent concentration compactness principle, we prove the existence of a nontrivial weak solution for sufficiently large $\lambda$, with the solution norm decaying to zero as $\lambda \to +\infty$. This extends subcritical results to the variable critical framework.

1. Introduction

This paper is devoted to investigating the existence of nontrivial solutions for a new $p\left(x\right)$-Kirchhoff problem with critical growth:

$$\left\{\begin{array}{c}-\left(a-b{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right){\Delta }_{p\left(x\right)}u=\lambda f(x,u)+{\left|u\right|}^{q\left(x\right)-2}u,\mathrm{in}\Omega ,\hfill \\ u=0\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(1)

where $a,b>0$ are constants; $\lambda >0$ is a parameter; ${\Delta }_{p\left(x\right)}u:={\mathrm{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is the $p\left(x\right)$-Laplacian operator; and $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\partial \Omega$. We assume that $p\left(x\right),q\left(x\right)$ are continuous functions on $\overline{\Omega }$ and satisfy the following conditions:

$$\begin{array}{c}2p_{-}>p_{+},1<p_{-}\le p_{+}<N,\\ p_{+}<q\left(x\right)\le p^{*}\left(x\right)\mathrm{for}x\in \overline{\Omega }\mathrm{and}\{x\in \Omega \mid q\left(x\right)=p^{*}\left(x\right)\}\ne \varnothing ,\end{array}$$

(2)

where $p_{-}:=\underset{\Omega }{inf}p\left(x\right)$, $p_{+}:=\underset{\Omega }{sup}p\left(x\right)$ and $p^{*}\left(x\right)={\displaystyle \frac{Np\left(x\right)}{N-p\left(x\right)}}$. Similarly, one can define $q_{-},q_{+}$. The nonlinear term f is continuous on $\overline{\Omega }\times \mathbb{R}$ and satisfies the following conditions:

($f_1$)

there exists $s\in (p_{+},N)$ such that

$$\underset{t\to 0}{lim}{\displaystyle \frac{f(x,t)}{{\left|t\right|}^{s-1}}}=0\mathrm{uniformly}\mathrm{in}\Omega ;$$

($f_2$)

there exists $r\in (p_{+},q_{-})$ such that

$$\underset{\left|t\right|\to +\infty }{lim}{\displaystyle \frac{f(x,t)}{{\left|t\right|}^{r-1}}}=0\mathrm{uniformly}\mathrm{in}\Omega ;$$

($f_3$)

there exists $\theta \in (p_{+},q_{-})$ such that for any $t\ne 0$ and any $x\in \Omega$, it holds

$$0<\theta F(x,t)\le tf(x,t),$$

where $F(x,t)={\int }_0^{t}f(x,s)\mathrm{d}s$.

Conditions $\left(f_1\right)$–$\left(f_2\right)$ show f is the subcritical growth. Condition $\left(f_3\right)$ is well known as the Ambrosetti–Rabinowitz super-linear condition, and it plays an important role in proving the boundedness of the Palais–Smale (PS) sequence corresponding to the Euler–Lagrange functional.

In view of the presence of ${\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x$, Equation (1) is called a nonlocal equation, which means that Equation (1) is no longer pointwise identity. These nonlocal equations originated from a model introduced by Kirchhoff in 1883:

$$\rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\left({\displaystyle \frac{P_0}{h}}+{\displaystyle \frac{E}{2L}}{\int }_0^L{\left|{\displaystyle \frac{\partial u}{\partial x}}\right|}^2\mathrm{d}x\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0,$$

which took into account the effect arising from change in the length of the string during the transverse vibration. Hence, these nonlocal equations are also referred to as Kirchhoff type equations.

The $p\left(x\right)$-Laplacian operator can be applied to fields such as image processing, quantum mechanics, elastic mechanics and electrorheological fluids. In the last few decades, the $p\left(x\right)$-Laplacian equation

$$\left\{\begin{array}{cc}-{\Delta }_{p\left(x\right)}u=g(x,u)\hfill & \hfill \mathrm{in}\Omega ,\\ u=0\hfill & \hfill \mathrm{on}\partial \Omega \end{array}\right.$$

(3)

has attracted considerable attention. There have been some significant research results concerning the $p\left(x\right)$-Laplacian equation. For example, Fan et al. [1] studied the eigenvalues of problem (3) when $g(x,u)={\lambda \left|u\right|}^{p\left(x\right)-2}u$ and gave some sufficient conditions for ensuring that the first eigenvalue of the $p\left(x\right)$-Laplacian operator is positive. In ref. [2], the authors studied the existence of nodal radial solutions for (3) again in the case where $g(x,u)={\left|u\right|}^{q\left(x\right)-2}u-a\left(x\right){\left|u\right|}^{p\left(x\right)-2}u$. In ref. [3], by applying the concentration compactness method for variable exponents, the authors proved the existence of multiple solutions for Equation (3) when $g(x,u)={\left|u\right|}^{q\left(x\right)-2}u+\lambda f(x,u)$ and generalized the previous results to the variable exponent case. In ref. [4], the authors considered the case where $g(x,u)=f(x,u)+{\lambda \left|u\right|}^{q\left(x\right)-2}u$ and $q\left(x\right)$ is a critical variable exponent. They proved the existence of multiple solutions to Equation (3) for $\lambda$ small enough. Up to now, the $p\left(x\right)$-Kirchhoff type equations have been extensively studied. In ref. [5], the authors dealt with the following problem:

$$\left\{\begin{array}{cc}-\left(a+b{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right){\Delta }_{p\left(x\right)}u=g(x,u)\hfill & \hfill \mathrm{in}\Omega ,\\ u=0\hfill & \hfill \mathrm{on}\partial \Omega \end{array}\right.$$

(4)

and proved the existence of infinitely many positive solutions to Equation (4) with g satisfying some certain conditions. Dai and Wei [6] proved the existence of a sequence of non-zero non-negative weak solutions to (4) under different hypotheses. In ref. [7], the authors investigated the following problem:

$$\left\{\begin{array}{cc}-M\left(𝒜\left(u\right)\right)div\left({a\left(\right|\nabla u|}^{p\left(x\right)}{\left)\right|\nabla u|}^{p\left(x\right)-2}\nabla u\right)=\lambda f(x,u)+{\left|u\right|}^{s\left(x\right)-2}u\hfill & \hfill \mathrm{in}\Omega ,\\ u=0\hfill & \hfill \mathrm{on}\partial \Omega ,\end{array}\right.$$

(5)

where $𝒜\left(u\right)={\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{A\left(\right|\nabla u|}^{p\left(x\right)})\mathrm{d}x$, f satisfies suitable conditions, and $s\left(x\right)$ is a critical variable exponent. They obtained the existence and the asymptotic behavior of the nontrivial weak solutions to (5) for $\lambda$ large enough. For more research findings on the $p\left(x\right)$-Kirchhoff type equations, consult references [8,9,10,11,12].

Recently, a new class of $p\left(x\right)$-Kirchhoff type equations has been attracting increasing attention, characterized by the presence of a negative nonlocal term. Such equations typically take the following form:

$$\left\{\begin{array}{cc}-\left(a-b{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right){\Delta }_{p\left(x\right)}u=f(x,u)\hfill & \hfill \mathrm{in}\Omega ,\\ u=0\hfill & \hfill \mathrm{on}\partial \Omega .\end{array}\right.$$

(6)

In ref. [13], the authors studied Equation (6) when $f(x,u)={\lambda \left|u\right|}^{p\left(x\right)-2}u+g(x,u)$, where g satisfies the Ambrosetti–Rabinowitz’s superlinear condition. They proved the existence and multiplicity results for (6). Chu and Liu [14] obtained the existence and multiplicity of solutions to (6) in the case where $f(x,u)={\left|u\right|}^{p_{+}-2}u$ with the help of the perturbation technique. In ref. [15], the authors investigated the same problem as in [13], and under weaker assumptions on the nonlinear term g, they demonstrated the existence of infinitely many solutions. Especially, we note that a recent work by Carrero and Hernández-Llanos [16] also treats a Kirchhoff-type equation with a negative nonlocal structure, though in the setting of fractional Laplacian operators.

While existing works have obtained valuable results for negative nonlocal $p\left(x\right)$-Kirchhoff equations, they are confined to subcritical nonlinearities, and the corresponding variational techniques fail in the variable critical setting. This paper studies a modified $p\left(x\right)$-Kirchhoff problem (1) with a negative nonlocal coefficient and variable critical growth, a model distinct from previously studied critical Kirchhoff equations such as (5). The structural difference and loss of compactness create new analytical challenges, which we resolve via a tailored variational framework. Our formulation therefore presents a genuinely new problem whose analysis is mathematically meaningful and worthy of rigorous investigation.

The Euler–Lagrange functional $J_{\lambda }:W_0^{1,p(·)}(\Omega )\to \mathbb{R}$ corresponding to (1) is defined by

$$\begin{array}{cc}\hfill J_{\lambda }\left(u\right)=& a{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x-{\displaystyle \frac{b}{2}}{\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right)}^2\hfill \\ & -\lambda {\int }_{\Omega }F(x,u)\mathrm{d}x-{\int }_{\Omega }{\displaystyle \frac{1}{q\left(x\right)}}{\left|u\right|}^{q\left(x\right)}\mathrm{d}x.\hfill \end{array}$$

(7)

It is easy to see that $J_{\lambda }$ is $C^1$, and for any $u,\phi \in W_0^{1,p(·)}(\Omega )$,

$$\begin{array}{cc}\hfill ⟨J_{\lambda }^{\prime }\left(u\right),\phi ⟩=& a{\int }_{\Omega }{|\nabla u|}^{p\left(x\right)-2}\nabla u\nabla \phi \mathrm{d}x-b\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right)\left({\int }_{\Omega }{|\nabla u|}^{p\left(x\right)-2}\nabla u\nabla \phi \mathrm{d}x\right)\hfill \\ & -\lambda {\int }_{\Omega }f(x,u)\phi \mathrm{d}x-{\int }_{\Omega }{\left|u\right|}^{q\left(x\right)-2}u\phi \mathrm{d}x.\hfill \end{array}$$

It is easy to check that a nontrivial critical point of $J_{\lambda }$ is a nontrivial solution to (1). Our main result is the following:

Theorem 1.

Suppose that $p,q\in C\left(\overline{\Omega }\right)$ satisfy condition (2) and $\left(f_1\right)$–$\left(f_3\right)$ hold. Then there exists ${\lambda }_{*}>0$ such that for $\lambda \ge {\lambda }_{*}$, Equation (1) admits at least a nontrivial solution $u_{\lambda }$. Additionally, $\underset{\lambda \to +\infty }{lim}\parallel u_{\lambda }\parallel =0$.

Remark 1.

The presence of the nonlocal term introduces difficulties in proving the boundedness of the (PS) sequence. Meanwhile, the critical variable exponent leads to a lack of compactness. To overcome these issues, we employ a truncation technique. By applying the concentration compactness principle for variable exponents, we first establish the existence of the nontrivial critical point for the truncated functional. Subsequently, we demonstrate that for λ large enough, the critical point of the truncated functional becomes that of the original functional.

Throughout this article, $B_R\left(x\right)$ denotes by set $\{y\in {\mathbb{R}}^N:|y-x|<R\}$ and $|\Omega |$ denotes the Lebesgue measure of the set $\Omega$. C, $C_i$($i=1,2\cdots$) stand for positive constants. The paper is organized as follows. In Section 2, we present some preliminary results. In Section 3, we prove the Palais–Smale condition of the truncated functional. In Section 4, we prove the main result.

2. Preliminary Results

In this section, we will give the definitions of variable exponent Lebesgue–Sobolev spaces and some conclusions about them. Define $C_{+}\left(\overline{\Omega }\right)=\{h\left(x\right)\in C\left(\overline{\Omega }\right)|h\left(x\right)>1\mathrm{for}\mathrm{all}x\in \overline{\Omega }\}$. For $h\in C_{+}\left(\overline{\Omega }\right)$, one denotes $h_{-}=\underset{\Omega }{inf}h\left(x\right)$ and $h_{+}=\underset{\Omega }{sup}h\left(x\right)$. The variable exponent Lebesgue space $L^{h(·)}(\Omega )$ is defined by

$$L^{h(·)}(\Omega )=\left\{u\right|u\mathrm{is}\mathrm{a}\mathrm{measurable}\mathrm{real}-\mathrm{valued}\mathrm{function}\mathrm{and}{\int }_{\Omega }{\left|u\right|}^{h\left(x\right)}\mathrm{d}x<+\infty \}$$

with the norm

$${\parallel u\parallel }_{h(·)}=inf\left\{\mu >0:{\int }_{\Omega }{\left|{\displaystyle \frac{u}{\mu }}\right|}^{h\left(x\right)}\mathrm{d}x\le 1\right\}.$$

The following lemma establishes the relationship between ${\int }_{\Omega }{\left|u\right|}^{h\left(x\right)}\mathrm{d}x$ and ${\parallel u\parallel }_{h(·)}$.

Lemma 1

([17] Proposition 2.3). Let ${\rho }_{h(·)}\left(u\right):={\int }_{\Omega }{\left|u\right|}^{h\left(x\right)}\mathrm{d}x$. For $u\in L^{h(·)}(\Omega )$,

(1) 

if ${\parallel u\parallel }_{h(·)}>1$, then ${\parallel u\parallel }_{h(·)}^{h_{-}}\le {\rho }_{h(·)}\left(u\right)\le {\parallel u\parallel }_{h(·)}^{h_{+}}$;

(2) 

if ${\parallel u\parallel }_{h(·)}<1$, then ${\parallel u\parallel }_{h(·)}^{h_{+}}\le {\rho }_{h(·)}\left(u\right)\le {\parallel u\parallel }_{h(·)}^{h_{-}}$.

The variable exponent Sobolev space $W^{1,h(·)}(\Omega )$ is defined by

$$W^{1,h(·)}(\Omega )=\{u\in L^{h(·)}(\Omega )\left|\right|\nabla u|\in L^{h(·)}(\Omega )\},$$

which is endowed with the norm

$${\parallel u\parallel }_{W^{1,h(·)}(\Omega )}={\parallel u\parallel }_{h(·)}+{\parallel \nabla u\parallel }_{h(·)}.$$

Define $W_0^{1,h(·)}(\Omega )$ as the closure of $C_c^{\infty }(\Omega )$ in $W^{1,h(·)}(\Omega )$. According to Propositions 2.1 and 2.5 in [17], the spaces $L^{h(·)}(\Omega )$, $W^{1,h(·)}(\Omega )$ and $W_0^{1,h(·)}(\Omega )$ are separable and reflexive Banach spaces if $1<h_{-}\le h_{+}<\infty$.

Next, we introduce some fundamental properties of variable exponent Lebesgue–Sobolev spaces. Analogous to classical Lebesgue–Sobolev spaces, these variable exponent spaces also satisfy corresponding Poincaré inequalities, Sobolev embeddings and Hölder inequalities.

Lemma 2

([13] Proposition 2.3). For $h\in C_{+}\left(\overline{\Omega }\right)$, there is a constant $C>0$ such that for all $u\in W_0^{1,h(·)}(\Omega )$,

$${\parallel u\parallel }_{h(·)}\le C{\parallel \nabla u\parallel }_{h(·)}.$$

Remark 2.

According to Lemma 2, ${\parallel u\parallel }_{W^{1,h(·)}(\Omega )}$ and ${\parallel \nabla u\parallel }_{h(·)}$ are the equivalent norms on $W_0^{1,h(·)}(\Omega )$. Therefore, we define $\parallel u\parallel :={\parallel \nabla u\parallel }_{h(·)}$ as the norm on $W_0^{1,h(·)}(\Omega )$.

Lemma 3

([13] Proposition 2.2). For $p,h\in C_{+}\left(\overline{\Omega }\right)$ satisfying $h\left(x\right)\le p^{*}\left(x\right)$ for any $x\in \overline{\Omega }$, there is a continuous embedding $W^{1,p(·)}(\Omega )\hookrightarrow L^{h(·)}(\Omega )$. Furthermore, if $h\left(x\right)<p^{*}\left(x\right)$ for any $x\in \overline{\Omega }$, the embedding is compact.

Lemma 4

([18] Proposition 2.1).

(i) 

For $h\in C_{+}\left(\overline{\Omega }\right)$, let $L^{d(·)}(\Omega )$ be the dual space of $L^{h(·)}(\Omega )$, where ${\displaystyle \frac{1}{h\left(x\right)}}+{\displaystyle \frac{1}{d\left(x\right)}}=1$. For $u\in L^{h(·)}(\Omega )$ and $v\in L^{d(·)}(\Omega )$, it holds

$$\left|{\int }_{\Omega }uv\mathrm{d}x\right|\le \left({\displaystyle \frac{1}{h_{-}}}+{\displaystyle \frac{1}{d_{-}}}\right){\parallel u\parallel }_{h(·)}{\parallel v\parallel }_{d(·)}.$$

(ii) 

For $h_1,h_2\in C_{+}\left(\overline{\Omega }\right)$ satisfying $h_1\left(x\right)\le h_2\left(x\right)$ for $x\in \overline{\Omega }$, one has $L^{h_2(·)}(\Omega )\hookrightarrow L^{h_1(·)}(\Omega )$.

Lemma 5

([15] Proposition 2.2). Suppose that $h\in L^{\infty }(\Omega )$ satisfies ${ess\; inf}_{x\in \Omega }h\left(x\right)>1$ and $p\in C_{+}\left(\overline{\Omega }\right)$. If ${\left|u\right|}^{h\left(x\right)}\in L^{p(·)}(\Omega )$, then it holds that

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

Analogous to the constant exponent case, the Brézis–Lieb lemma corresponding to the variable exponent case also holds.

Lemma 6

([19] Lemma 2.1). Assume that $1<h_{-}\le h\left(x\right)\le h_{+}<N$. If $\left\{u_n\right\}$ is bounded in $L^{h(·)}(\Omega )$ and $u_n\to u\in L^{h(·)}(\Omega )$ a.e. on Ω, then

$$\underset{n\to +\infty }{lim}{\int }_{\Omega }\left(|u_n{|}^{h\left(x\right)}-{|u_n-u|}^{h\left(x\right)}\right)\mathrm{d}x={\int }_{\Omega }{\left|u\right|}^{h\left(x\right)}\mathrm{d}x.$$

The following lemma plays a crucial role in verifying the Palais–Smale (PS) condition.

Lemma 7

([17] Theorem 3.1). For $p\in C_{+}\left(\overline{\Omega }\right)$, define

$$L\left(u\right):={\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}xforu\in W_0^{1,p(·)}(\Omega ).$$

Then, the following conclusions hold:

• $L\in C^1(W_0^{1,p(·)}(\Omega ),\mathbb{R})$, and the derivative operator $L^{\prime }$ is

  $$⟨L^{\prime }\left(u\right),v⟩={\int }_{\Omega }{|\nabla u|}^{p\left(x\right)-2}\nabla u\nabla v\mathrm{d}xforallu,v\in W_0^{1,p(·)}(\Omega ).$$
• If $u_n\rightharpoonup u$ as $n\to +\infty$ and $\underset{n\to +\infty }{lim\; sup}⟨L^{\prime }\left(u_n\right),u_n-u⟩\le 0$, then $u_n\to u$ strongly in $W_0^{1,p(·)}(\Omega )$ as $n\to +\infty$.

3. The Palais–Smale Condition of the Truncated Functional

In this section, we construct a truncated functional and establish that it satisfies the Palais–Smale condition under specific assumptions.

Take a cut-off function $\varphi \in C^1\left([0,+\infty ),\mathbb{R}\right)$ satisfying $0\le \varphi \le 1$, ${\varphi }^{\prime }\le 0$, $\underset{t>0}{max}\left|{\varphi }^{\prime }\left(t\right)\right|\le 2$ and

$$\varphi \left(t\right)=\left\{\begin{array}{c}1,\mathrm{if}0\le t\le 1,\hfill \\ 0,\mathrm{if}t\ge 2.\hfill \end{array}\right.$$

Define $S:=\underset{\phi \in C_c^{\infty }(\Omega )}{inf}{\displaystyle \frac{{\parallel \nabla \phi \parallel }_{p(·)}}{{\parallel \phi \parallel }_{q(·)}}}$. Let $K:={\left[min\left\{{\left({\displaystyle \frac{a}{2}}\right)}^{{\displaystyle \frac{1}{p_{-}}}},{\left({\displaystyle \frac{a}{2}}\right)}^{{\displaystyle \frac{1}{p_{+}}}}\right\}S\right]}^N$. For

$$0<T<min\left\{{\displaystyle \frac{a}{4b}},{\left({\displaystyle \frac{(q_{-}-\theta )K}{2q_{-}(2b\theta +8p_{+}b)}}\right)}^{{\displaystyle \frac{1}{2}}},{\displaystyle \frac{a(\theta -p_{+})}{2b\theta +8p_{+}b}}\right\}.$$

(8)

Remark on the choice of T. The upper bound $T<{\displaystyle \frac{a(\theta -p_{+})}{2b\theta +8b{p}_{+}}}$ is chosen to ensure the positivity of the energy estimate in Lemma 13. Indeed, if $L\left(u\right)\ge T$, then using $\varphi \le 1$, ${\varphi }^{\prime }\ge -2$ and $L\left(u\right)\le 2T$ whenever the truncated term is active, we obtain

$$\begin{array}{cc}\hfill J_{\lambda ,T}\left(u\right)-{\displaystyle \frac{1}{\theta }}⟨J_{\lambda ,T}^{\prime }\left(u\right),u⟩& \ge a\left(1-{\displaystyle \frac{p_{+}}{\theta }}\right)L\left(u\right)-{\displaystyle \frac{b}{2}}{\left(2T\right)}^2-{\displaystyle \frac{8b{p}_{+}}{\theta }}{T}^2\hfill \\ & \ge a\left(1-{\displaystyle \frac{p_{+}}{\theta }}\right)T-\left(2b+{\displaystyle \frac{8b{p}_{+}}{\theta }}\right)T^2.\hfill \end{array}$$

The right-hand side is strictly positive precisely when $T<{\displaystyle \frac{a(\theta -p_{+})}{2b\theta +8b{p}_{+}}}$. This positivity is essential for the contradiction argument in Lemma 13.

We consider the truncated functional $J_{\lambda ,T}:W_0^{1,p(·)}(\Omega )\to \mathbb{R}$, defined by

$$\begin{array}{cc}\hfill J_{\lambda ,T}\left(u\right)=& a{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x-{\displaystyle \frac{b}{2}}\varphi \left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x}{T}}\right){\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right)}^2\hfill \\ & -\lambda {\int }_{\Omega }F(x,u)\mathrm{d}x-{\int }_{\Omega }{\displaystyle \frac{1}{q\left(x\right)}}{\left|u\right|}^{q\left(x\right)}\mathrm{d}x.\hfill \end{array}$$

For $u\in W_0^{1,p(·)}(\Omega )$, it has

$$\begin{array}{cc}\hfill & ⟨J_{\lambda ,T}^{\prime }\left(u\right),\phi ⟩\hfill \\ \hfill =& a{\int }_{\Omega }{|\nabla u|}^{p\left(x\right)-2}\nabla u\nabla \phi \mathrm{d}x\hfill \\ & -{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x}{T}}\right){\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right)}^2{\int }_{\Omega }{|\nabla u|}^{p\left(x\right)-2}\nabla u\nabla \phi \mathrm{d}x\hfill \\ & -b\varphi \left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x}{T}}\right)\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right){\int }_{\Omega }{|\nabla u|}^{p\left(x\right)-2}\nabla u\nabla \phi \mathrm{d}x\hfill \\ & -\lambda {\int }_{\Omega }f(x,u)\phi \mathrm{d}x-{\int }_{\Omega }{\left|u\right|}^{q\left(x\right)-2}u\phi \mathrm{d}x.\hfill \end{array}$$

Next, we will verify that the truncated functional $J_{\lambda ,T}$ has the mountain pass geometry.

Lemma 8.

Suppose that $\left(2\right)$ and $\left(f_1\right)$–$\left(f_2\right)$ hold. Then there exists $\alpha ,\rho >0$ such that for any $u\in W_0^{1,p(·)}(\Omega )$ satisfying $\parallel u\parallel =\rho$, it has $J_{\lambda ,T}\left(u\right)\ge \alpha >0$.

Proof. 

Since $\left(f_1\right)$–$\left(f_2\right)$ hold, for any $\sigma >0$, one can choose $C\left(\sigma \right)>0$ such that

$$F(x,t)\le {\displaystyle \frac{\sigma }{s}}{\left|t\right|}^s+{\displaystyle \frac{C\left(\sigma \right)}{r}}{\left|t\right|}^r\mathrm{for}\mathrm{any}(x,t)\in \overline{\Omega }\times \mathbb{R}.$$

Then, by Lemmas 1 and 3, one has

$$\begin{array}{cc}\hfill J_{\lambda ,T}\left(u\right)\ge & a{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x-{\displaystyle \frac{b}{2}}{\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right)}^2-{\displaystyle \frac{\lambda \sigma }{s}}{\int }_{\Omega }{\left|u\right|}^s\mathrm{d}x\hfill \\ & -{\displaystyle \frac{\lambda C\left(\sigma \right)}{r}}{\int }_{\Omega }{\left|u\right|}^r\mathrm{d}x-{\int }_{\Omega }{\displaystyle \frac{1}{q\left(x\right)}}{\left|u\right|}^{q\left(x\right)}\mathrm{d}x\hfill \\ \hfill \ge & a{\displaystyle \frac{1}{p_{+}}}{\int }_{\Omega }{|\nabla u|}^{p\left(x\right)}\mathrm{d}x-{\displaystyle \frac{b}{2p_{-}^2}}{\left({\int }_{\Omega }{|\nabla u|}^{p\left(x\right)}\mathrm{d}x\right)}^2-{\displaystyle \frac{\lambda \sigma C_1}{s}}{\parallel u\parallel }^s-{\displaystyle \frac{\lambda C\left(\sigma \right)C_2}{r}}{\parallel u\parallel }^r\hfill \\ & -{\displaystyle \frac{1}{q_{-}}}max\left\{{\left|u\right|}_{q\left(x\right)}^{q_{-}},{\left|u\right|}_{q\left(x\right)}^{q_{+}}\right\}\hfill \\ \hfill \ge & {\displaystyle \frac{a}{p_{+}}}{\parallel u\parallel }^{p_{+}}-{\displaystyle \frac{b}{2p_{-}^2}}{\parallel u\parallel }^{2p_{-}}-{\displaystyle \frac{\lambda \sigma C_1}{s}}{\parallel u\parallel }^s-{\displaystyle \frac{\lambda C\left(\sigma \right)C_2}{r}}{\parallel u\parallel }^r-{\displaystyle \frac{C_3}{q_{-}}}max\left\{{\parallel u\parallel }^{q_{-}},{\parallel u\parallel }^{q_{+}}\right\},\hfill \end{array}$$

(9)

where we assume $\parallel u\parallel ={|\nabla u|}_{p(·)}\le 1$. Since $2p_{-},s,r,q_{-}>p_{+}$, there exist $\rho ,\alpha >0$ such that $J_{\lambda ,T}\left(u\right)\ge \alpha >0$ for $\parallel u\parallel =\rho$. □

Lemma 9.

Suppose that $\left(f_3\right)$ holds. Then there exists a nonnegative function $u_{*}\in W_0^{1,p(·)}(\Omega )$ such that $\parallel u_{*}\parallel >\rho$ and $J_{\lambda ,T}\left(u_{*}\right)<0$.

Proof. 

By condition $\left(f_3\right)$, there exist positive constants $d_1,d_2$ such that $F(x,t)\ge d_1{t}^{\theta }-d_2$ for all $t>0$ and $x\in \Omega$. Take a nonnegative function $e\in C_c^{\infty }(\Omega )\setminus \left\{\theta \right\}$. By Lemma 1, one finds that for $t>0$,

$$\begin{array}{cc}\hfill J_{\lambda ,T}\left(te\right)& \le a{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla \left(te\right)|}^{p\left(x\right)}\mathrm{d}x-\lambda {\int }_{\Omega }\left(d_1{\left(te\right)}^{\theta }-d_2\right)\mathrm{d}x-{\int }_{\Omega }{\displaystyle \frac{1}{q\left(x\right)}}{\left|te\right|}^{q\left(x\right)}\mathrm{d}x\hfill \\ & \le {\displaystyle \frac{a}{p_{-}}}max\{{\parallel te\parallel }^{p_{-}},{\parallel te\parallel }^{p_{+}}\}-\lambda d_1{\int }_{\Omega }{\left(te\right)}^{\theta }\mathrm{d}x+\lambda {\int }_{\Omega }{d}_2\mathrm{d}x-{\displaystyle \frac{1}{q_{+}}}{\int }_{\Omega }{\left|te\right|}^{q\left(x\right)}\mathrm{d}x\hfill \\ & \le {\displaystyle \frac{a}{p_{-}}}max\{{\parallel te\parallel }^{p_{-}},{\parallel te\parallel }^{p_{+}}\}-\lambda d_1{\int }_{\Omega }{\left(te\right)}^{\theta }\mathrm{d}x+\lambda d_2|\Omega |-{\displaystyle \frac{1}{q_{+}}}min\left\{{\parallel te\parallel }_{q(·)}^{q_{-}},{\parallel te\parallel }_{q(·)}^{q_{+}}\right\}.\hfill \end{array}$$

Since $p_{+}<\theta <q_{-}$, one can see that $J_{\lambda ,T}\left(te\right)\to -\infty$ as $t\to +\infty$. Therefore, one chooses $t_{*}>0$ such that $J_{\lambda ,T}\left(t_{*}e\right)<0$ and $\parallel t_{*}e\parallel >\rho$. By taking $u_{*}=t_{*}e$, the proof is completed. □

Let $c_{\lambda ,T}:=\underset{\gamma \in \Gamma }{inf}\underset{t\in [0,1]}{max}{J}_{\lambda ,T}\left(\gamma \left(t\right)\right)$, where

$$\Gamma =\{\gamma \in C\left([0,1],W_0^{1,p(·)}(\Omega )\right)|\gamma \left(0\right)=0,J_{\lambda ,T}\left(\gamma \left(1\right)\right)<0\}.$$

According to Lemma 8, it has $c_{\lambda ,T}\ge \alpha >0$.

Next, we investigate how the mountain pass energy level $c_{\lambda ,T}$ behaves as the parameter $\lambda$ varies.

Lemma 10.

Suppose that $\left(2\right)$ and $\left(f_1\right)$–$\left(f_3\right)$ hold. Then $\underset{\lambda \to +\infty }{lim}{c}_{\lambda ,T}=0$.

Proof. 

Let e be that taken in the proof of Lemma 9. For $t>0$, let $g\left(t\right)=J_{\lambda ,T}\left(te\right)$. By computation, one obtains

$$\begin{array}{cc}\hfill g^{\prime }\left(t\right)=& a{\int }_{\Omega }{t}^{p\left(x\right)-1}{|\nabla e|}^{p\left(x\right)}\mathrm{d}x\hfill \\ & -{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla \left(te\right)|}^{p\left(x\right)}\mathrm{d}x}{T}}\right)\left({\int }_{\Omega }{t}^{p\left(x\right)-1}{|\nabla e|}^{p\left(x\right)}\mathrm{d}x\right){\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla \left(te\right)|}^{p\left(x\right)}\mathrm{d}x\right)}^2\hfill \\ & -b\varphi \left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla \left(te\right)|}^{p\left(x\right)}\mathrm{d}x}{T}}\right)\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla \left(te\right)|}^{p\left(x\right)}\mathrm{d}x\right)\left({\int }_{\Omega }{t}^{p\left(x\right)-1}{|\nabla e|}^{p\left(x\right)}\mathrm{d}x\right)\hfill \\ & -\lambda {\int }_{\Omega }f(x,te)e\mathrm{d}x-{\int }_{\Omega }{t}^{q\left(x\right)-1}{\left|e\right|}^{q\left(x\right)}\mathrm{d}x.\hfill \end{array}$$

Since $J_{\lambda ,T}\left(te\right)\to -\infty$ as $t\to +\infty$, by Lemma 8, there exists $t_{\lambda }>0$ such that $g\left(t_{\lambda }\right)=\underset{t\ge 0}{max}g\left(t\right)$. Thus, $g^{\prime }\left(t_{\lambda }\right)=0$, i.e.,

$$\begin{array}{cc}\hfill & a⟨L^{\prime }\left(t_{\lambda }e\right),t_{\lambda }e⟩-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left({\displaystyle \frac{L\left(t_{\lambda }e\right)}{T}}\right)⟨L^{\prime }\left(t_{\lambda }e\right),t_{\lambda }e⟩{\left(L\left(t_{\lambda }e\right)\right)}^2\hfill \\ & -b\varphi \left({\displaystyle \frac{L\left(t_{\lambda }e\right)}{T}}\right)L\left(t_{\lambda }e\right)⟨L^{\prime }\left(t_{\lambda }e\right),t_{\lambda }e⟩=\lambda {\int }_{\Omega }f(x,t_{\lambda }e)t_{\lambda }e\mathrm{d}x+{\int }_{\Omega }{\left|t_{\lambda }e\right|}^{q\left(x\right)}\mathrm{d}x,\hfill \end{array}$$

(10)

where L is defined in Lemma 7. By the definition of $\varphi$ and Lemma 1, the left-hand side of (10) admits the following estimate:

$$\begin{array}{cc}\hfill & a⟨L^{\prime }\left(t_{\lambda }e\right),t_{\lambda }e⟩-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left({\displaystyle \frac{L\left(t_{\lambda }e\right)}{T}}\right)⟨L^{\prime }\left(t_{\lambda }e\right),t_{\lambda }e⟩L{\left(t_{\lambda }e\right)}^2-b\varphi \left({\displaystyle \frac{L\left(t_{\lambda }e\right)}{T}}\right)L\left(t_{\lambda }e\right)⟨L^{\prime }\left(t_{\lambda }e\right),t_{\lambda }e⟩\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le & amax\{\parallel t_{\lambda }{e\parallel }^{p_{-}},\parallel t_{\lambda }e{\parallel }^{p_{+}}\}+8b{p}_{+}{T}^2.\hfill \end{array}$$

Applying Lemma 1 again, one has ${\int }_{\Omega }{|t_{\lambda }e|}^{q\left(x\right)}\mathrm{d}x\ge min\{\parallel t_{\lambda }{e\parallel }_{q(·)}^{q_{-}},\parallel t_{\lambda }e{\parallel }_{q(·)}^{q_{+}}\}$. Since $p_{+}<q_{-}$, by (10), one can deduce ${\left\{t_{\lambda }\right\}}_{\lambda }$ is bounded. Now choose a monotonic sequence $\left\{{\lambda }_n\right\}$ such that ${\lambda }_n\to +\infty$ and $t_{{\lambda }_n}\to t_0$ as $n\to +\infty$. Suppose $t_0>0$. By (10) again,

$$\begin{array}{cc}\hfill & a⟨L^{\prime }\left(t_{{\lambda }_n}e\right),t_{{\lambda }_n}e⟩-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left({\displaystyle \frac{L\left(t_{{\lambda }_n}e\right)}{T}}\right)⟨L^{\prime }\left(t_{{\lambda }_n}e\right),t_{{\lambda }_n}e⟩{\left(L\left(t_{{\lambda }_n}e\right)\right)}^2\hfill \\ & -b\varphi \left({\displaystyle \frac{L\left(t_{{\lambda }_n}e\right)}{T}}\right)L\left(t_{{\lambda }_n}e\right)⟨L^{\prime }\left(t_{{\lambda }_n}e\right),t_{{\lambda }_n}e⟩={\lambda }_n{\int }_{\Omega }f(x,t_{{\lambda }_n}e)t_{{\lambda }_n}e\mathrm{d}x+{\int }_{\Omega }{\left|t_{{\lambda }_n}e\right|}^{q\left(x\right)}\mathrm{d}x.\hfill \end{array}$$

(11)

Letting $n\to +\infty$ in Equation (11), the left-hand side is bounded while the right-hand side is unbounded, which is impossible. Hence, $t_0=0$. By the definition of $c_{\lambda ,T}$, one has

$$c_{{\lambda }_n,T}\le \underset{0\le t\le 1}{max}{J}_{{\lambda }_n,T}\left(t{t}_{*}e\right)\le J_{{\lambda }_n,T}\left(t_{{\lambda }_n}e\right)\le a{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla \left(t_{{\lambda }_n}e\right)|}^{p\left(x\right)}\mathrm{d}x\to 0$$

as $n\to +\infty$. By the definition of $c_{\lambda ,T}$ again, it is easy to see that $c_{\lambda ,T}$ is monotonically decreasing in $\lambda$. Hence, one obtains $\underset{\lambda \to +\infty }{lim}{c}_{\lambda ,T}=0$. □

The following result represents the central lemma of this paper. Given that our study focuses on the nonlocal equation with variable exponents, the proof of this lemma becomes particularly complex.

Lemma 11.

Suppose that $\left(f_1\right)$–$\left(f_3\right)$ hold. Then $J_{\lambda ,T}$ satisfies the ${\left(PS\right)}_c$ condition for $c<{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{q_{-}}}\right)K$, where $K:={\left[min\left\{{({\displaystyle \frac{a}{2}})}^{{\displaystyle \frac{1}{p_{-}}}},{({\displaystyle \frac{a}{2}})}^{{\displaystyle \frac{1}{p_{+}}}}\right\}S\right]}^N$.

Proof. 

Let $\left\{u_n\right\}\subset W_0^{1,p(·)}(\Omega )$ be a ${\left(PS\right)}_c$ sequence, i.e.,

$$J_{\lambda ,T}\left(u_n\right)\to c\mathrm{and}{J}_{\lambda ,T}^{\prime }\left(u_n\right)\to 0\mathrm{in}{W}_0^{1,p(·)}(\Omega )\mathrm{as}n\to +\infty .$$

The proof is divided into three parts.

Part 1: Proving$\left\{u_n\right\}$ is bounded in $W_0^{1,p(·)}(\Omega )$.

By the definition of $\varphi$ and condition $\left(f_3\right)$, it holds

$$\begin{array}{cc}\hfill & c+1+o\left(1\right)\parallel u_n\parallel \hfill \\ \hfill \ge & J_{\lambda ,T}\left(u_n\right)-{\displaystyle \frac{1}{\theta }}⟨J_{\lambda ,T}^{\prime }\left(u_n\right),u_n⟩\hfill \\ \hfill \ge & \left(a-{\displaystyle \frac{a{p}_{+}}{\theta }}\right)L\left(u_n\right)-{\displaystyle \frac{b}{2}}\varphi \left({\displaystyle \frac{L\left(u_n\right)}{T}}\right){\left(L\left(u_n\right)\right)}^2+{\displaystyle \frac{b}{2T\theta }}{\varphi }^{\prime }\left({\displaystyle \frac{L\left(u_n\right)}{T}}\right){\left(L\left(u_n\right)\right)}^2⟨L^{\prime }\left(u_n\right),u_n⟩\hfill \\ & +{\displaystyle \frac{b}{\theta }}\varphi \left({\displaystyle \frac{L\left(u_n\right)}{T}}\right)L\left(u_n\right)⟨L^{\prime }\left(u_n\right),u_n⟩+{\displaystyle \frac{\lambda }{\theta }}{\int }_{\Omega }f(x,u_n)u_n\mathrm{d}x-\lambda {\int }_{\Omega }F(x,u_n)\mathrm{d}x\hfill \\ & +{\displaystyle \frac{1}{\theta }}{\int }_{\Omega }|u_n{|}^{q\left(x\right)}\mathrm{d}x-{\int }_{\Omega }{\displaystyle \frac{1}{q\left(x\right)}}{\left|u_n\right|}^{q\left(x\right)}\mathrm{d}x\hfill \\ \hfill \ge & a\left(1-{\displaystyle \frac{p_{+}}{\theta }}\right)L\left(u_n\right)-2b{T}^2-{\displaystyle \frac{8b{T}^2{p}_{+}}{\theta }}\hfill \\ \hfill \ge & {\displaystyle \frac{a}{p_{+}}}\left(1-{\displaystyle \frac{p_{+}}{\theta }}\right)min\{{\parallel u_n\parallel }^{p_{-}},{\parallel u_n\parallel }^{p_{+}}\}-2b{T}^2-{\displaystyle \frac{8b{T}^2{p}_{+}}{\theta }}.\hfill \end{array}$$

(12)

Hence, $\left\{u_n\right\}$ is bounded in $W_0^{1,p(·)}(\Omega )$. Since $W_0^{1,p(·)}(\Omega )$ is a reflexive Banach space, there exists a subsequence still denoted by $\left\{u_n\right\}$ and $u\in W_0^{1,p(·)}(\Omega )$ such that as $n\to +\infty$,

$$\begin{array}{cc}\hfill u_n\rightharpoonup u& \mathrm{in}{W}_0^{1,p(·)}(\Omega ),\hfill \\ \hfill u_n\to u& \mathrm{in}{L}^{h(·)}(\Omega )\mathrm{for}h\in C_{+}\left(\overline{\Omega }\right)\mathrm{satisfying}h\left(x\right)<p^{*}\left(x\right)\mathrm{on}\overline{\Omega },\hfill \\ \hfill u_n\to u& \mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .\hfill \end{array}$$

(13)

Moreover, there exist nonnegative and bounded measures $\mu ,\nu$ on $\overline{\Omega }$ such that $|\nabla u_n{|}^{p\left(x\right)}\rightharpoonup \mu$, $|u_n{|}^{q\left(x\right)}\rightharpoonup \nu$ weakly in the sense of measures. According to the concentration compactness principle for the variable exponents ([4] Lemma 3.4), there exists a countable index set I, ${\left\{x_i\right\}}_{i\in I}\subset \{x\in \Omega :q\left(x\right)=p^{*}\left(x\right)\}\subset \Omega$ and ${\left\{{\mu }_i\right\}}_{i\in I},{\left\{{\nu }_i\right\}}_{i\in I}\subset [0,+\infty )$ such that

$$\begin{array}{c}\hfill \mu \ge {|\nabla u|}^{p\left(x\right)}\mathrm{d}x+\sum _{i\in I}{\mu }_i{\delta }_{x_i},{\mu }_i>0,\\ \hfill \nu ={\left|u\right|}^{q\left(x\right)}\mathrm{d}x+\sum _{i\in I}{\nu }_i{\delta }_{x_i},{\nu }_i>0,\\ \hfill S{\nu }_i^{{\displaystyle \frac{1}{p^{*}\left(x_i\right)}}}\le {\mu }_i^{{\displaystyle \frac{1}{p\left(x_i\right)}}},\mathrm{for}\mathrm{all}i\in I,\end{array}$$

where ${\delta }_{x_i}$ is the Dirac mass at $x_i$, and S is the best constant in the Gagliardo–Nirenberg–Sobolev inequality for variable exponents:

$$S=\underset{\phi \in C_c^{\infty }(\Omega )}{inf}{\displaystyle \frac{{\parallel \nabla \phi \parallel }_{p(·)}}{{\parallel \phi \parallel }_{q(·)}}}.$$

(14)

Part 2: Proving I is empty.

Suppose by contradiction that I is nonempty. Let $\eta \in C_c^{\infty }({\mathbb{R}}^N,[0,1])$, satisfying

$$\eta \left(x\right)=\left\{\begin{array}{c}\hfill 1,\left|x\right|<1;\\ \hfill 0,\left|x\right|\ge 2.\end{array}\right.$$

For $i\in I$ and any $ϵ>0$, set ${\eta }_{ϵ,i}:=\eta \left({\displaystyle \frac{x-x_i}{ϵ}}\right)$. Since $\left\{u_n\right\}$ is bounded in $W_0^{1,p(·)}(\Omega )$, one has

$$\begin{array}{cc}\hfill o\left(1\right)=& ⟨J_{\lambda ,T}^{\prime }\left(u_n\right),u_n{\eta }_{ϵ,i}⟩\hfill \\ \hfill =& \left[a-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left({\displaystyle \frac{L\left(u_n\right)}{T}}\right)L{\left(u_n\right)}^2-b\varphi \left({\displaystyle \frac{L\left(u_n\right)}{T}}\right)L\left(u_n\right)\right]·\hfill \\ & \left({\int }_{\Omega }{|\nabla u_n|}^{p\left(x\right)}{\eta }_{ϵ,i}\mathrm{d}x+{\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x\right)\hfill \\ & -\lambda {\int }_{\Omega }f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x-{\int }_{\Omega }{\left|u_n\right|}^{q\left(x\right)}{\eta }_{ϵ,i}\mathrm{d}x\hfill \\ \hfill \ge & (a-2bT){\int }_{\Omega }{|\nabla u_n|}^{p\left(x\right)}{\eta }_{ϵ,i}\mathrm{d}x+\left[a-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left({\displaystyle \frac{L\left(u_n\right)}{T}}\right)L{\left(u_n\right)}^2-b\varphi \left({\displaystyle \frac{L\left(u_n\right)}{T}}\right)L\left(u_n\right)\right]·\hfill \\ & {\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x-\lambda {\int }_{\Omega }f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x-{\int }_{\Omega }{\left|u_n\right|}^{q\left(x\right)}{\eta }_{ϵ,i}\mathrm{d}x\hfill \\ \hfill \ge & (a-2bT){\int }_{\Omega }{|\nabla u_n|}^{p\left(x\right)}{\eta }_{ϵ,i}\mathrm{d}x-(a+4bT)\left|{\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x\right|\hfill \\ & -\lambda {\int }_{\Omega }f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x-{\int }_{\Omega }{\left|u_n\right|}^{q\left(x\right)}{\eta }_{ϵ,i}\mathrm{d}x.\hfill \end{array}$$

(15)

The proof of Part 2 proceeds in three steps.

Step 1: Proving $\underset{ϵ\to 0}{lim}\underset{n\to +\infty }{lim\; sup}\left|{\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x\right|=0$. By Lemmas 4 and 5, one has

$$\begin{array}{cc}\hfill & \left|{\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x\right|\hfill \\ \hfill \le & {\int }_{\Omega }|u_n\left|\right|\nabla {\eta }_{ϵ,i}\left|\right|\nabla u_n{|}^{p\left(x\right)-1}\mathrm{d}x\hfill \\ \hfill \le & C_4{∥|u_n\left|\right|\nabla {\eta }_{ϵ,i}|∥}_{p(·)}{∥|\nabla u_n{|}^{p\left(x\right)-1}∥}_{p{\left(x\right)}^{\prime }}\hfill \\ \hfill \le & C_{4}max\left\{{\parallel \nabla u_n\parallel }_{p(·)}^{p_{-}-1},{\parallel \nabla u_n\parallel }_{p(·)}^{p_{+}-1}\right\}·\hfill \\ & max\left\{{\left({\int }_{\Omega }|u_n|\nabla {\eta }_{ϵ,i}|{|}^{p\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{1}{p_{-}}}},{\left({\int }_{\Omega }|u_n|\nabla {\eta }_{ϵ,i}|{|}^{p\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{1}{p_{+}}}}\right\}.\hfill \end{array}$$

(16)

By (13), $u_n\to u$ in $L^{p(·)}(\Omega )$ as $n\to +\infty$. Hence, with the help of Lemma 6, one can deduce that as $n\to +\infty$,

$${\int }_{\Omega }{\left|u_n|\nabla {\eta }_{ϵ,i}|\right|}^{p\left(x\right)}\mathrm{d}x\to {\int }_{\Omega }{\left|u|\nabla {\eta }_{ϵ,i}|\right|}^{p\left(x\right)}\mathrm{d}x.$$

(17)

Since $\left\{u_n\right\}$ is bounded in $W_0^{1,p(·)}(\Omega )$, combining (16) and (17), it follows that

$$\begin{array}{cc}\hfill & \left|{\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x\right|\hfill \\ \hfill \le & C_{5}max\left\{{\left({\int }_{\Omega }|u|\nabla {\eta }_{ϵ,i}|{|}^{p\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{1}{p_{-}}}},{\left({\int }_{\Omega }|u|\nabla {\eta }_{ϵ,i}|{|}^{p\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{1}{p_{+}}}}\right\}+o_n\left(1\right).\hfill \end{array}$$

(18)

By Lemmas 1 and 4, one finds

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\left|u\right|}^{p\left(x\right)}{|\nabla {\eta }_{ϵ,i}|}^{p\left(x\right)}\mathrm{d}x\hfill \\ \hfill \le & C_6{∥{\left|u\right|}^{p\left(x\right)}∥}_{L^{{\displaystyle \frac{N}{N-p(·)}}}\left(B_{2ϵ}\left(x_i\right)\right)}{∥|\nabla {\eta }_{ϵ,i}{|}^{p\left(x\right)}∥}_{L^{{\displaystyle \frac{N}{p(·)}}}\left(B_{2ϵ}\left(x_i\right)\right)}\hfill \\ \hfill \le & C_{6}max\left\{{\left({\int }_{B_{2ϵ}\left(x_i\right)}{\left|u\right|}^{p^{*}\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{N-p_{-}}{N}}},{\left({\int }_{B_{2ϵ}\left(x_i\right)}{\left|u\right|}^{p^{*}\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{N-p_{+}}{N}}}\right\}·\hfill \\ & max\left\{{\left({\int }_{B_{2ϵ}\left(x_i\right)}{|\nabla {\eta }_{ϵ,i}|}^N\mathrm{d}x\right)}^{{\displaystyle \frac{p_{+}}{N}}},{\left({\int }_{B_{2ϵ}\left(x_i\right)}{|\nabla {\eta }_{ϵ,i}|}^N\mathrm{d}x\right)}^{{\displaystyle \frac{p_{-}}{N}}}\right\}\hfill \\ \hfill \le & C_{7}max\left\{{\left({\int }_{B_{2ϵ}\left(x_i\right)}{\left|u\right|}^{p^{*}\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{N-p_{-}}{N}}},{\left({\int }_{B_{2ϵ}\left(x_i\right)}{\left|u\right|}^{p^{*}\left(x\right)}\mathrm{d}x\right)}^{{\displaystyle \frac{N-p_{+}}{N}}}\right\}.\hfill \end{array}$$

(19)

Combining (18) and (19), one obtains $\underset{ϵ\to 0}{lim}\underset{n\to +\infty }{lim\; sup}\left|{\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x\right|=0$.

Step 2: Proving $\underset{ϵ\to 0}{lim}\underset{n\to +\infty }{lim}{\int }_{\Omega }f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x=0$. Since $u_n$ is bounded in $W_0^{1,p(·)}(\Omega )$ and $W_0^{1,p(·)}(\Omega )\hookrightarrow L^r(\Omega )$, there exists $C_0>0$ such that ${\int }_{\Omega }{\left|u_n\right|}^r\mathrm{d}x\le C_0$ for all $n\in {\mathbb{N}}^{+}$. For any $\sigma >0$, by $\left(f_2\right)$, there exists $M\left(\sigma \right)>0$ such that

$$\left|f\right(x,t\left)t\right|\le M\left(\sigma \right)+{\displaystyle \frac{\sigma }{2C_0}}{\left|t\right|}^r$$

(20)

for $x\in \Omega$ and $t\in \mathbb{R}$. Take $\delta ={\displaystyle \frac{\sigma }{2M\left(\sigma \right)}}$. For $E\subset \Omega$ satisfying $meas\left(E\right)<\delta$, then one obtains

$$\begin{array}{cc}\hfill \left|{\int }_{E}f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x\right|& \le {\int }_E\left|f(x,u_n)u_n\right|\mathrm{d}x\hfill \\ & \le {\int }_{E}M\left(\sigma \right)\mathrm{d}x+{\int }_E{\displaystyle \frac{\sigma }{2C_0}}{\left|u_n\right|}^r\mathrm{d}x\hfill \\ & \le \delta M\left(\sigma \right)+{\displaystyle \frac{\sigma }{2C_0}}·C_0=\sigma .\hfill \end{array}$$

(21)

Hence, ${\left\{{\int }_{\Omega }f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x\right\}}_{n\in {\mathbb{N}}^{+}}$ is equi-absolutely continuous. By Vitali’s convergence theorem, it holds that

$$\underset{n\to +\infty }{lim}{\int }_{\Omega }f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x={\int }_{\Omega }f(x,u)u{\eta }_{ϵ,i}\mathrm{d}x.$$

(22)

Thus, $\underset{ϵ\to 0}{lim}\underset{n\to +\infty }{lim}{\int }_{\Omega }f(x,u_n)u_n{\eta }_{ϵ,i}\mathrm{d}x=0$.

Step 3: Proving I is empty. Letting $n\to +\infty$ in (15), it follows that

$$\begin{array}{cc}\hfill 0\ge & (a-2bT){\int }_{\Omega }{\eta }_{ϵ,i}\mathrm{d}\mu -\underset{n\to +\infty }{lim\; sup}(a+4bT)\left|{\int }_{\Omega }{u}_n{|\nabla u_n|}^{p\left(x\right)-2}\nabla u_n\nabla {\eta }_{ϵ,i}\mathrm{d}x\right|\hfill \\ & -\lambda {\int }_{\Omega }f(x,u)u{\eta }_{ϵ,i}\mathrm{d}x-{\int }_{\Omega }{\eta }_{ϵ,i}\mathrm{d}\nu .\hfill \end{array}$$

(23)

Letting $ϵ\to 0$ in (23), one has

$$0\ge (a-2bT){\mu }_i-{\nu }_i\ge {\displaystyle \frac{a}{2}}{\mu }_i-{\nu }_i$$

(24)

since T is defined in (8), naturally it satisfies $T<{\displaystyle \frac{a}{4b}}$. It follows that

$$\begin{array}{c}\hfill {\nu }_i\ge {\displaystyle \frac{a}{2}}{\mu }_i\ge {\displaystyle \frac{a}{2}}{\left(S{\nu }_i^{{\displaystyle \frac{1}{p^{*}\left(x_i\right)}}}\right)}^{p\left(x_i\right)}={\displaystyle \frac{a}{2}}{S}^{p\left(x_i\right)}{\nu }_i^{{\displaystyle \frac{N-p\left(x_i\right)}{N}}}.\end{array}$$

(25)

Hence,

$${\nu }_i\ge {\left[min\left\{{\left({\displaystyle \frac{a}{2}}\right)}^{{\displaystyle \frac{1}{p_{-}}}},{\left({\displaystyle \frac{a}{2}}\right)}^{{\displaystyle \frac{1}{p_{+}}}}\right\}S\right]}^N=K.$$

(26)

By $\left(f_3\right)$, it holds

$$\begin{array}{cc}\hfill c+o\left(1\right)=& J_{\lambda ,T}\left(u_n\right)-{\displaystyle \frac{1}{\theta }}⟨J_{\lambda ,T}^{\prime }\left(u_n\right),u_n⟩\hfill \\ \hfill =& a{\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_n|}^{p\left(x\right)}\mathrm{d}x-a{\int }_{\Omega }{\displaystyle \frac{1}{\theta }}{|\nabla u_n|}^{p\left(x\right)}\mathrm{d}x-{\displaystyle \frac{b}{2}}\varphi \left(L\left(u_n\right)\right)L{\left(u_n\right)}^2\hfill \\ & +{\displaystyle \frac{b}{2T\theta }}{\varphi }^{\prime }\left(L\left(u_n\right)\right)L{\left(u_n\right)}^2⟨L^{\prime }\left(u_n\right),u_n⟩+{\displaystyle \frac{b}{\theta }}\varphi \left(L\left(u_n\right)\right)L\left(u_n\right)⟨L^{\prime }\left(u_n\right),u_n⟩\hfill \\ & +{\displaystyle \frac{\lambda }{\theta }}{\int }_{\Omega }f(x,u_n)u_n\mathrm{d}x-\lambda {\int }_{\Omega }F(x,u_n)\mathrm{d}x+{\displaystyle \frac{1}{\theta }}{\int }_{\Omega }|u_n{|}^{q\left(x\right)}\mathrm{d}x-{\int }_{\Omega }{\displaystyle \frac{1}{q\left(x\right)}}{\left|u_n\right|}^{q\left(x\right)}\mathrm{d}x\hfill \\ \hfill \ge & -{\displaystyle \frac{b}{2}}·{\left(2T\right)}^2-{\displaystyle \frac{p_{+}b}{T\theta }}·{\left(2T\right)}^2·2T+\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{q_{-}}}\right){\int }_{\Omega }{\left|u_n\right|}^{q\left(x\right)}\mathrm{d}x\hfill \\ \hfill =& \left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{q_{-}}}\right){\int }_{\Omega }{\left|u_n\right|}^{q\left(x\right)}\mathrm{d}x-2b{T}^2-{\displaystyle \frac{8p_{+}b}{\theta }}·T^2\hfill \\ \hfill \ge & \left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{q_{-}}}\right){\nu }_i-2b{T}^2-{\displaystyle \frac{8p_{+}b}{\theta }}·T^2+o\left(1\right)\hfill \\ \hfill \ge & \left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{q_{-}}}\right)K-\left(2b+{\displaystyle \frac{8p_{+}b}{\theta }}\right)T^2+o\left(1\right)\hfill \\ \hfill \ge & {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{q_{-}}}\right)K+o\left(1\right),\hfill \end{array}$$

(27)

which is a contradiction. Therefore, I is empty, which implies that as $n\to +\infty$,

$${\int }_{\Omega }|u_n{|}^{q\left(x\right)}\mathrm{d}x\to {\int }_{\Omega }{\left|u\right|}^{q\left(x\right)}\mathrm{d}x.$$

(28)

Part 3: Proving$\underset{n\to +\infty }{lim}⟨L^{\prime }\left(u_n\right),(u_n-u)⟩=0$.

Applying Lemmas 1 and 6, one can deduce that $\parallel u_n{-u\parallel }_{q(·)}\to 0$ as $n\to +\infty$. By $\left(f_1\right)$–$\left(f_2\right)$ and (13), one has

$$\begin{array}{cc}\hfill & \left|{\int }_{\Omega }f(x,u_n)(u_n-u)\mathrm{d}x\right|\hfill \\ \hfill \le & {\int }_{\Omega }\left|f(x,u_n)(u_n-u)\right|\mathrm{d}x\hfill \\ \hfill \le & {\int }_{\Omega }\left(\sigma |u_n{|}^{s-1}+C\left(\sigma \right){\left|u_n\right|}^{r-1}\right)|u_n-u|\mathrm{d}x\hfill \\ \hfill \le & \sigma {\left({\int }_{\Omega }{\left|u_n\right|}^s\mathrm{d}x\right)}^{{\displaystyle \frac{s-1}{s}}}{\left({\int }_{\Omega }{|u_n-u|}^s\mathrm{d}x\right)}^{{\displaystyle \frac{1}{s}}}+C\left(\sigma \right){\left({\int }_{\Omega }{\left|u_n\right|}^r\mathrm{d}x\right)}^{{\displaystyle \frac{r-1}{r}}}{\left({\int }_{\Omega }{|u_n-u|}^r\mathrm{d}x\right)}^{{\displaystyle \frac{1}{r}}}\hfill \\ \hfill \le & C_8{\left({\int }_{\Omega }{|u_n-u|}^s\mathrm{d}x\right)}^{{\displaystyle \frac{1}{s}}}+C_9{\left({\int }_{\Omega }{|u_n-u|}^r\mathrm{d}x\right)}^{{\displaystyle \frac{1}{r}}}\hfill \\ \hfill \to & 0\hfill \end{array}$$

(29)

as $n\to +\infty$. Moreover, by Lemma 4,

$$\begin{array}{cc}\hfill \left|{\int }_{\Omega }{\left|u_n\right|}^{q\left(x\right)-2}{u}_n(u_n-u)\mathrm{d}x\right|\le & {\int }_{\Omega }|u_n{|}^{q\left(x\right)-1}|u_n-u|\mathrm{d}x\hfill \\ \hfill \le & C_{10}{∥{\left|u_n\right|}^{q\left(x\right)-1}∥}_{{\displaystyle \frac{q(·)}{q(·)-1}}}{\parallel u_n-u\parallel }_{q(·)}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to & 0\hfill \end{array}$$

as $n\to +\infty$. It follows that

$$\begin{array}{cc}\hfill o\left(1\right)=& ⟨J_{\lambda ,T}^{\prime }\left(u_n\right),u_n-u⟩\hfill \\ \hfill =& ⟨L^{\prime }\left(u_n\right),(u_n-u)⟩\left[a-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left(L\left(u_n\right)\right)L{\left(u_n\right)}^2-b\varphi \left(L\left(u_n\right)\right)L\left(u_n\right)\right]\hfill \\ & -\lambda {\int }_{\Omega }f(x,u_n)(u_n-u)\mathrm{d}x-{\int }_{\Omega }{\left|u_n\right|}^{q\left(x\right)-2}{u}_n(u_n-u)\mathrm{d}x.\hfill \end{array}$$

(30)

Hence, $\underset{n\to +\infty }{lim}⟨L^{\prime }\left(u_n\right),(u_n-u)⟩\left[a-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left(L\left(u_n\right)\right)L{\left(u_n\right)}^2-b\varphi \left(L\left(u_n\right)\right)L\left(u_n\right)\right]=0$. Since ${\varphi }^{\prime }\le 0$ and $\varphi \ge 0$, the nonlocal terms in the factor are nonnegative; hence, the factor is at least a and, in particular, at least $a-2bT$ (which is positive since $T<{\displaystyle \frac{a}{4b}}$). Thus,

$$\begin{array}{cc}\hfill & \left|⟨L^{\prime }\left(u_n\right),(u_n-u)⟩\left[a-{\displaystyle \frac{b}{2T}}{\varphi }^{\prime }\left(L\left(u_n\right)\right)L{\left(u_n\right)}^2-b\varphi \left(L\left(u_n\right)\right)L\left(u_n\right)\right]\right|\hfill \\ \hfill \ge & \left|⟨L^{\prime }\left(u_n\right),(u_n-u)⟩\right|(a-2bT)\ge {\displaystyle \frac{a}{2}}\left|⟨L^{\prime }\left(u_n\right),(u_n-u)⟩\right|\ge 0,\hfill \end{array}$$

(31)

it holds that $\underset{n\to +\infty }{lim}⟨L^{\prime }\left(u_n\right),(u_n-u)⟩=0$. By Lemma 7, one obtains $u_n\to u$ in $W_0^{1,p(·)}(\Omega )$. The proof is completed. □

4. Existence of the Nontrivial Solution

In this section, we establish the main result by first demonstrating that the truncated functional admits a nontrivial critical point.

Lemma 12.

Suppose that $\left(2\right)$ and $\left(f_1\right)$–$\left(f_3\right)$ hold. Then there exists $\overline{\lambda }>0$ such that for $\lambda \ge \overline{\lambda }$, $J_{\lambda ,T}$ has a nontrivial critical point $u_{\lambda }$.

Proof. 

According to Lemmas 10, there exists $\overline{\lambda }>0$ such that $c_{\lambda ,T}<{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{q_{-}}}\right)K$ for $\lambda \ge \overline{\lambda }$. Combining Lemmas 8 and 9 with the mountain pass theorem without the (PS) condition (Theorem 2.2 in [20]), there exists a sequence ${\left\{u_n\right\}}_{n\ge 1}\subset W_0^{1,p(·)}(\Omega )$ satisfying

$$J_{\lambda ,T}\left(u_n\right)\to c_{\lambda ,T}\mathrm{and}{J}_{\lambda ,T}^{\prime }\left(u_n\right)\to 0\mathrm{in}{\left(W_0^{1,p(·)}(\Omega )\right)}^{*}$$

as $n\to +\infty$. For $\lambda \ge \overline{\lambda }$, by Lemma 11, there exist a subsequence $\left\{u_{n_k}\right\}$ and $u_{\lambda }\in W_0^{1,p(·)}(\Omega )$ such that $u_{n_k}\to u_{\lambda }$ in $W_0^{1,p(·)}(\Omega )$ as $k\to +\infty$. It follows that

$$J_{\lambda ,T}\left(u_{\lambda }\right)=c_{\lambda ,T}>0\mathrm{and}{J}_{\lambda ,T}^{\prime }\left(u_{\lambda }\right)=0.$$

(32)

Therefore, $u_{\lambda }$ is a nontrivial critical point of $J_{\lambda ,T}$ for $\lambda \ge \overline{\lambda }$. □

To ensure that the critical point of the truncated functional $J_{\lambda ,T}$ corresponds to that of the original functional $J_{\lambda }$, we require the following result.

Lemma 13.

There exists ${\lambda }_{*}\ge \overline{\lambda }>0$ such that if $\lambda \ge {\lambda }_{*}$, then ${\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_{\lambda }|}^{p\left(x\right)}\mathrm{d}x<T$, where $u_{\lambda }$ is obtained in Lemma 12.

Proof. 

Suppose by contradiction that there exists a sequence $\left\{{\lambda }_n\right\}$ satisfying ${\lambda }_n\ge \overline{\lambda }$ for all $n\in {\mathbb{N}}^{+}$ and ${\lambda }_n\to +\infty$ as $n\to +\infty$ such that ${\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x\ge T$. Since $T<{\displaystyle \frac{a(\theta -p_{+})}{2\theta b+8b{p}_{+}}}$, recall that this upper bound is exactly the one chosen in the definition of T; see the remark after the definition of T in Section 3. It follows that

$$\begin{array}{cc}\hfill & c_{{\lambda }_n,T}\hfill \\ \hfill =& J_{{\lambda }_n,T}\left(u_{{\lambda }_n}\right)-{\displaystyle \frac{1}{\theta }}⟨J_{{\lambda }_n,T}^{\prime }\left(u_{{\lambda }_n}\right),u_{{\lambda }_n}⟩\hfill \\ \hfill \ge & \left(a-{\displaystyle \frac{a{p}_{+}}{\theta }}\right){\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x-{\displaystyle \frac{b}{2}}\varphi \left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x}{T}}\right){\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x\right)}^2\hfill \\ & +{\displaystyle \frac{b}{2T\theta }}{\varphi }^{\prime }\left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x}{T}}\right){\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x\right)}^2{\int }_{\Omega }{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x\hfill \\ & +{\displaystyle \frac{b}{\theta }}\varphi \left({\displaystyle \frac{{\int }_{\Omega }\frac{1}{p\left(x\right)}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x}{T}}\right)\left({\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x\right)·\left({\int }_{\Omega }{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x\right)\hfill \\ & +{\displaystyle \frac{{\lambda }_n}{\theta }}{\int }_{\Omega }f(x,u_{{\lambda }_n})u_{{\lambda }_n}\mathrm{d}x-{\lambda }_n{\int }_{\Omega }F(x,u_{{\lambda }_n})\mathrm{d}x+{\displaystyle \frac{1}{\theta }}{\int }_{\Omega }|u_{{\lambda }_n}{|}^{q\left(x\right)}\mathrm{d}x-{\int }_{\Omega }{\displaystyle \frac{1}{q\left(x\right)}}{\left|u_{{\lambda }_n}\right|}^{q\left(x\right)}\mathrm{d}x\hfill \\ \hfill \ge & a\left(1-{\displaystyle \frac{p_{+}}{\theta }}\right){\int }_{\Omega }{\displaystyle \frac{1}{p\left(x\right)}}{|\nabla u_{{\lambda }_n}|}^{p\left(x\right)}\mathrm{d}x-2b{T}^2-{\displaystyle \frac{8b{T}^2{p}_{+}}{\theta }}\hfill \\ \hfill \ge & a\left(1-{\displaystyle \frac{p_{+}}{\theta }}\right)T-\left(2b+{\displaystyle \frac{8b{p}_{+}}{\theta }}\right)T^2>0,\hfill \end{array}$$

(33)

which contradicts with $\underset{n\to +\infty }{lim}{c}_{{\lambda }_n,T}=0$. The proof is completed. □

The Proof of Theorem 1. 

By Lemmas 12 and 13, for $\lambda \ge {\lambda }_{*}$, one can see that $u_{\lambda }$ is also a nontrivial critical point of $J_{\lambda }$. Thus, $u_{\lambda }$ is a nontrivial solution of problem (1).

It remains to prove the decay property. By Lemma 13, for $\lambda \ge {\lambda }_{*}$, we have $L\left(u_{\lambda }\right)<T$. Since $J_{\lambda ,T}\left(u_{\lambda }\right)=c_{\lambda ,T}\to 0$ as $\lambda \to +\infty$ (Lemma 10), the coercivity estimate

$$J_{\lambda ,T}\left(u\right)\ge {\displaystyle \frac{a}{p_{+}}}{\parallel u\parallel }^{p_{+}}-{\displaystyle \frac{b}{2p_{-}^2}}{\parallel u\parallel }^{2p_{-}}-{\displaystyle \frac{\lambda \sigma C_1}{s}}{\parallel u\parallel }^s-{\displaystyle \frac{\lambda C\left(\sigma \right)C_2}{r}}{\parallel u\parallel }^r-{\displaystyle \frac{C_3}{q_{-}}}max\left\{{\parallel u\parallel }^{q_{-}},{\parallel u\parallel }^{q_{+}}\right\},$$

(9) implies that $\parallel u_{\lambda }\parallel \to 0$. Indeed, if $\parallel u_{\lambda }\parallel$ did not tend to zero, the positive leading term ${\displaystyle \frac{a}{p_{+}}}{\parallel u_{\lambda }\parallel }^{p_{+}}$ in (9) would prevent $J_{\lambda ,T}\left(u_{\lambda }\right)$ from approaching zero. Therefore,

$$\underset{\lambda \to +\infty }{lim}\parallel u_{\lambda }\parallel =0.$$

The proof is complete. □