Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept

Abstract

Motivated by the idea which has been introduced by Boulaaras and Guefaifia (Math. Meth. Appl. Sci. 41 (2018), 5203–5210 and, by Afrouzi and Shakeri (Afr. Mat. (2015) 26:159–168) combined with some properties of Kirchhoff type operators, we prove the existence of positive solutions for a class of nonlocal $p\left(x\right)$-Kirchhoff evolutionary systems by using the sub and super solutions concept.

1. Introduction

The study of differential equations and variational problems with nonstandard $p\left(x\right)$-growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [1,2]). Many existence results have been obtained on this kind of problems—see, for example, [3,4,5,6]. In Refs. [7,8,9,10,11,12], Fan et al. studied the regularity of solutions for differential equations with nonstandard $p\left(x\right)$-growth conditions.

In this article, we are interested in the $p\left(x\right)$-Kirchhoff parabolic systems of the form

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}}-M\left(I_0\left(u\right)\right){\Delta }_{p\left(x\right)}u={\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u\right)\right]\mathrm{in}{Q}_T=\mathsf{\Omega }\times \left[0,T\right],\hfill \\ {\displaystyle \frac{\partial u}{\partial t}}-M\left(I_0\left(v\right)\right){\Delta }_{p\left(x\right)}v={\lambda }^{p\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]\mathrm{in}{Q}_T=\mathsf{\Omega }\times \left[0,T\right],\hfill \\ u=v=0\mathrm{on}\partial Q_T,\hfill \\ u(x,0)=\phi \left(x\right),\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth domain with $C^2$ boundary $\partial \mathsf{\Omega },1<p\left(x\right)\in C^1\left(\overline{\mathsf{\Omega }}\right)$ is a functions with $1<p^{-}:={inf}_{\mathsf{\Omega }}p\left(x\right)\le p^{+}:={sup}_{\mathsf{\Omega }}p\left(x\right)<\infty ,{\Delta }_{p\left(x\right)}u=÷\left({\left|\nabla u\right|}^{p\left(x\right)-2}\nabla u\right)$ is called $p\left(x\right)$-Laplacian, $\lambda ,{\lambda }_1,{\lambda }_2,{\mu }_1,$ and ${\mu }_2$ are positive parameters, $I_0\left(u\right)=\underset{\mathsf{\Omega }}{\int }\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}dx$ and $M\left(t\right)$ is a continuous function.

Problem (1) is a generalization of a model introduced by Kirchhoff [13]. More precisely, Kirchhoff proposed a model given by the equation

$$\rho \frac{{\partial }^{2}u}{\partial t^2}-\left(\frac{P_0}{h}+\frac{E}{2L}\underset{0}{\overset{L}{\int }}{\left|\frac{\partial u}{\partial x}\right|}^{2}dx\right)\frac{{\partial }^{2}u}{\partial x^2}=0,$$

(2)

where $\rho$, $P_0$, $h,E,L$ are constants, which extends the classical D’Alembert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to [14,15,16,17,18,19,20,21], in which the authors have used a variational method and topological method to get the existence of solutions.

In this paper, motivated by the ideas introduced in [22] and the properties of Kirchhoff type operators in [22], we study the existence of positive solutions for system (2) by using the sub- and super solutions techniques. To our best knowledge, this is a new research topic for nonlocal problems. The remainder of this paper is organized as follows. In Section 2, we present some preleminary results on the variable exponent Sobolev space $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ and the method of sub- and super solutions. Section 3 is devoted to stating and proving the main result.

2. Preliminary Results

In order to discuss problem (1), we need some theories on $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ which we call variable exponent Sobolev space. Firstly, we state some basic properties of spaces $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ which will be used later (for details, see [3]).

Let us define

$$L^{p\left(x\right)}\left(\mathsf{\Omega }\right)=\left\{u:u\mathrm{is}\mathrm{a}\mathrm{measurable}\mathrm{real}-\mathrm{valued}\mathrm{function}\mathrm{such}\mathrm{that}\underset{\mathsf{\Omega }}{\int }{\left|u\left(x\right)\right|}^{p\left(x\right)}dx<\infty \right\}.$$

We introduce the norm on $L^{p\left(x\right)}\left(\mathsf{\Omega }\right)$ by

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

and

$$W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)=\left\{u\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right);\left|\nabla u\right|\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right)\right\},$$

with the norm

$$∥u∥={\left|u\right|}_{p\left(x\right)}+{\left|\nabla u\right|}_{p\left(x\right)},\forall u\in W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right).$$

We denote by $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ the closure of $C_0^{\infty }\left(\mathsf{\Omega }\right)$ in $W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right).$

Now, using Euler time scheme of problem (1), we obtain the following problems:

$$\left\{\begin{array}{c}{u}_k-{\tau }^{\prime }M\left(I_0\left(u_k\right)\right){\Delta }_{p\left(x\right)}{u}_k={\lambda }^{p\left(x\right)}{\tau }^{\prime }\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]+u_{k-1}\mathrm{in}\mathsf{\Omega },\hfill \\ u_k-{\tau }^{\prime }M\left(I_0\left(v\right)\right){\Delta }_{p\left(x\right)}v={\lambda }^{p\left(x\right)}{\tau }^{\prime }\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]+u_{k-1}\mathrm{in}\mathsf{\Omega },\hfill \\ u_k=v=0\mathrm{on}\partial \mathsf{\Omega },\hfill \\ u_0={\phi }_0,\hfill \end{array}\right.$$

(3)

where $N{\tau }^{\prime }=T,$ $0<{\tau }^{\prime }<1,$ and for $1\le k\le N.$

Proposition 1

(see [12]). The spaces $L^{p\left(x\right)}\left(\mathsf{\Omega }\right),W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ and $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ are separable and reflexive Banach spaces.

Throughout the paper, we will assume that:

Hypothesis 1 (H1).

$M:[0,+\infty )\to \left[m_0,m_{\infty }\right]$ is a continuous and increasing function with $m_0>0;$

Hypothesis 2 (H2).

$p\in C^1\left(\overline{\mathsf{\Omega }}\right)$ and $1<p^{-}\le p^{+};$

Hypothesis 3 (H3).

$f,g,h,\tau :\left[0,+\infty \right[\to \mathbb{R}$ are $C^1,$ monotone functions such that

$$\underset{u_k\to +\infty }{lim}f\left(u_k\right)=+\infty ,\underset{u_k\to +\infty }{lim}g\left(u_k\right)=+\infty ,\underset{u_k\to +\infty }{lim}h\left(u_k\right)=+\infty ,\underset{u_k\to +\infty }{lim}\tau \left(u_k\right)=+\infty ,$$

Hypothesis 4 (H4).

${lim}_{u_k\to +\infty }\frac{f\left(L{\left(g\left(u_k\right)\right)}^{\frac{1}{p^{-}-1}}\right)}{u_k^{p^{-}-1}}=0,$ for all $L>0,$

Hypothesis 5 (H5).

${lim}_{u_k\to +\infty }\frac{h\left(u\right)}{u_k^{p^{-}-1}}=0,$ and ${lim}_{u\to +\infty }\frac{\tau \left(u_k\right)}{u_k^{p^{-}-1}}=0.$

Hypothesis 6 (H6).

$a,b,c,d$$:\overline{\mathsf{\Omega }}\to \left(0,+\infty \right)$ are continuous functions, such that

$$\begin{array}{ccc}\hfill a_1& =& \underset{x\in \overline{\mathsf{\Omega }}}{min}a\left(x\right),b_1=\underset{x\in \overline{\mathsf{\Omega }}}{min}b\left(x\right),c_1=\underset{x\in \overline{\mathsf{\Omega }}}{min}c\left(x\right),d_1=\underset{x\in \overline{\mathsf{\Omega }}}{min}d\left(x\right),\hfill \\ \hfill a_2& =& \underset{x\in \overline{\mathsf{\Omega }}}{max}a\left(x\right),b_2=\underset{x\in \overline{\mathsf{\Omega }}}{max}b\left(x\right),c_2=\underset{x\in \overline{\mathsf{\Omega }}}{max}c\left(x\right),d_2=\underset{x\in \overline{\mathsf{\Omega }}}{max}d\left(x\right).\hfill \end{array}$$

Definition 1.

If $u_k,v\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$, we say that

$$-M\left(I_0\left(u_k\right)\right){\Delta }_{p\left(x\right)}{u}_k\le -M\left(I_0\left(v\right)\right){\Delta }_{p\left(x\right)}v$$

if, for all $\phi \in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ with $\phi \ge 0,$

$$M\left(I_0\left(u\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k·\nabla \phi dx\le M\left(I_0\left(v\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla v\right|}^{p\left(x\right)-2}\nabla v·\nabla \phi dx,$$

(4)

where

$$I_0\left(u_k\right)=\underset{\mathsf{\Omega }}{\int }\frac{1}{p\left(x\right)}{\left|\nabla u_k\right|}^{p\left(x\right)}dx.$$

Definition 2.

$\left(1\right)$ If $u_k,v\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$, $\left(u_k,v\right)$ is called a weak solution of the problem defined in (3) if it satisfies

$$\left\{\begin{array}{c}M\left(I_0\left(u_k\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k·\nabla \phi dx=\underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]-{\displaystyle \frac{u_k-u_{k-1}}{{\tau }^{\prime }}}\right]\phi dx,\\ M\left(I_0\left(v\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla v\right|}^{p\left(x\right)-2}\nabla v·\nabla \phi dx=\underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]-{\displaystyle \frac{u_k-u_{k-1}}{{\tau }^{\prime }}}\right]\phi dx,\end{array}\right.$$

for all $\phi \in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ with $\phi \ge 0.$

$\left(2\right)$ We say that $(u,v)$ is called a sub solution (respectively a super solution) of the problem defined in (3) if

$$\left\{\begin{array}{c}M\left(I_0\left(u_k\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k·\nabla \phi dx\le \left(respectively\ge \right)\hfill \\ \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]-{\displaystyle \frac{u_k-u_{k-1}}{{\tau }^{\prime }}}\right]\phi dx,\hfill \\ M\left(I_0\left(v\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla v\right|}^{p\left(x\right)-2}\nabla v·\nabla \phi dx\le \left(respectively\ge \right)\hfill \\ \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]-{\displaystyle \frac{u_k-u_{k-1}}{{\tau }^{\prime }}}\right]\phi dx,\hfill \end{array}\right.$$

Lemma 1

(see [22] comparison principle). Let $u_k,v\in W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ and (H1) hold. If

$$-M\left(I_0\left(u_k\right)\right){\Delta }_{p\left(x\right)}u\le -M\left(I_0\left(v\right)\right){\Delta }_{p\left(x\right)}v$$

and ${\left(u_k-v\right)}^{+}\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right),$ then $u_k\le v$ in $\mathsf{\Omega }.$

Lemma 2

(see [22]). Let (H1) hold. $\eta >0$ and let u be the unique solution of the problem

$$-\underset{\mathsf{\Omega }}{\int }M\left(I_0\left(u_k\right)\right)÷\left({\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k\right)=\mu \mathrm{in}\mathsf{\Omega }.$$

Set $h=\frac{m_0{p}^{-}}{2{\left|\mathsf{\Omega }\right|}^{\frac{1}{N}}{C}_0}.$ Then, when

$$\mu \ge h,{\left|u_k\right|}_{\infty }\le C^{*}{\mu }^{\frac{1}{p^{-}-1}},$$

and when

$$\mu <h,{\left|u_k\right|}_{\infty }\le C_{*}{\mu }^{\frac{1}{p^{+}-1}},$$

where $C^{*}$ and $C_{*}$ are positive constants depending $p^{+},p^{-},N,\left|\mathsf{\Omega }\right|,C_0$ and $m_0.$

Here and hereafter, we will use the notation $d\left(x,\partial \mathsf{\Omega }\right)$ to denote the distance of $x\in \mathsf{\Omega }$ to denote the distance of $\mathsf{\Omega }.$ Denote $d\left(x\right)=d\left(x,\partial \mathsf{\Omega }\right)$ and

$$\partial {\mathsf{\Omega }}_{\epsilon }=\left\{x\in \mathsf{\Omega }:d\left(x,\partial \mathsf{\Omega }\right)<\epsilon \right\}.$$

Since $\partial \mathsf{\Omega }$ is $C^2$ regularly, there exists a constant $\delta \in \left(0,1\right)$ such that $d\left(x\right)\in C^2\left(\overline{\partial {\mathsf{\Omega }}_{3\delta }}\right)$ and $\left|\nabla d\left(x\right)\right|=1.$

Denote

$$v_1\left(x\right)=\left\{\begin{array}{c}\gamma d\left(x\right),d\left(x\right)<\delta ,\hfill \\ \gamma \delta +\underset{\delta }{\overset{d\left(x\right)}{\int }}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)}^{\frac{2}{p^{-}-1}}dt,\delta \le d\left(x\right)<2\delta ,\hfill \\ \gamma \delta +\underset{\delta }{\overset{2\delta }{\int }}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{\left({\lambda }_1{b}_1+{\mu }_1{d}_1\right)}^{\frac{2}{p^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

$$v_2\left(x\right)=\left\{\begin{array}{c}\gamma d\left(x\right),d\left(x\right)<\delta ,\hfill \\ \gamma \delta +\underset{\delta }{\overset{d\left(x\right)}{\int }}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{q^{-}-1}}{\left({\lambda }_2{a}_2+{\mu }_2{c}_2\right)}^{\frac{2}{q^{-}-1}}dt,\delta \le d\left(x\right)<2\delta ,\hfill \\ \gamma \delta +\underset{\delta }{\overset{2\delta }{\int }}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{q^{-}-1}}{\left({\lambda }_2{b}_2+{\mu }_2{d}_2\right)}^{\frac{2}{q^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

Obviously, $0\le v_1\left(x\right),v_2\left(x\right)\in C^1\left(\overline{\mathsf{\Omega }}\right).$ Considering

$$\left\{\begin{array}{c}-M\left(\underset{\mathsf{\Omega }}{\int }\frac{1}{p\left(x\right)}{\left|\nabla u_k\right|}^{p\left(x\right)}dx\right){\Delta }_{p\left(x\right)}\omega \left(x\right)=\eta \mathrm{in}\mathsf{\Omega },\hfill \\ u=0\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(5)

we have the following Lemma:

Lemma 3

(see [23]). If positive parameter η is large enough and ω is the unique solution of (5), then we have

(i) 

For any$\theta \in \left(0,1\right),$there exists a positive constant$C_1$such that

$$C_1{\eta }^{\frac{1}{p^{+}-1+\theta }}\le \underset{x\in \overline{\mathsf{\Omega }}}{max}\omega \left(x\right).$$

(ii) 

There exists a positive constant $C_2$ such that

$$\underset{x\in \overline{\mathsf{\Omega }}}{max}\omega \left(x\right)\le C_2{\eta }^{\frac{1}{p^{-}-1}}.$$

3. Main Result

In the following, when there is no misunderstanding, we always use $C_i$ to denote positive constants.

Theorem 1.

Assume that the conditions (H1)–(H6) are satisfied. Then, problem (3) has a positive solution when λ is large enough.

Proof. 

We shall establish Theorem 1 by constructing a positive subsolution $\left({\varphi }_k,{\varphi }_1\right)$ and supersolution $\left(z_k,z_1\right)$ of the problem defined in (1) such that ${\varphi }_k\le z_k$ and ${\varphi }_1\le z_1.$ That is, $\left({\varphi }_k,{\varphi }_1\right)$ and $\left(z_k,z_1\right)$ satisfies

$$\left\{\begin{array}{c}M\left(I_0\left({\varphi }_k\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla {\varphi }_k\right|}^{p\left(x\right)-2}\nabla {\varphi }_k·\nabla qdx\le \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left({\varphi }_1\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_k\right)\right]-{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}}\right]qdx,\\ M\left(I_0\left({\varphi }_1\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla {\mathsf{\Phi }}_1\right|}^{p\left(x\right)-2}\nabla {\varphi }_1·\nabla qdx\le \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left({\varphi }_k\right)+{\mu }_{2}d\left(x\right)\tau \left({\varphi }_1\right)\right]-{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}}\right]qdx,\end{array}\right.$$

and

$$\left\{\begin{array}{c}M\left(I_0\left(z_k\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_k\right|}^{p\left(x\right)-2}\nabla z_k·\nabla qdx\ge \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(z_1\right)+{\mu }_{1}c\left(x\right)h\left(z_k\right)\right]-{\displaystyle \frac{z_k-z_{k-1}}{{\tau }^{\prime }}}\right]qdx,\\ M\left(I_0\left(z_1\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_1\right|}^{p\left(x\right)-2}\nabla z_1·\nabla qdx\ge \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(z_k\right)+{\mu }_{2}d\left(x\right)\tau \left(z_1\right)\right]-{\displaystyle \frac{z_k-z_{k-1}}{{\tau }^{\prime }}}\right]qdx,\end{array}\right.$$

for all $q\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ with $q\ge 0.$ According to the sub-super solution method for $p\left(x\right)$-Kirchhoff type equations (see [22]), then the problem defined in (1) has a positive solution.

Step 1. We will construct a subsolution of (1). Let $\sigma \in (0,\delta )$ be small enough.

Denote

$${\varphi }_k\left(x\right)=\left\{\begin{array}{c}{e}^{k^{\prime }d\left(x\right)}-1,d\left(x\right)<\sigma ,\hfill \\ e^{k^{\prime }\sigma }-1+\underset{\sigma }{\overset{d\left(x\right)}{\int }}{k}^{\prime }{e}^{k^{\prime }\sigma }{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}{\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)}^{\frac{2}{p^{-}-1}}dt,\sigma \le d\left(x\right)<2\delta ,\hfill \\ e^{k^{\prime }\sigma }-1+\underset{\sigma }{\overset{2\delta }{\int }}{k}^{\prime }{e}^{k^{\prime }\sigma }{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}{\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)}^{\frac{2}{p^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

$${\varphi }_1\left(x\right)=\left\{\begin{array}{c}{e}^{k^{\prime }d\left(x\right)}-1,d\left(x\right)<\sigma ,\hfill \\ e^{k^{\prime }\sigma }-1+\underset{\sigma }{\overset{d\left(x\right)}{\int }}{k}^{\prime }{e}^{k^{\prime }\sigma }{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{q^{-}-1}}{\left({\lambda }_2{b}_1+{\mu }_2{d}_1\right)}^{\frac{2}{q^{-}-1}}dt,\sigma \le d\left(x\right)<2\delta ,\hfill \\ e^{k^{\prime }\sigma }-1+\underset{\sigma }{\overset{2\delta }{\int }}{k}^{\prime }{e}^{k^{\prime }\sigma }{\left(\frac{2\delta -t}{2\delta -\sigma }\right)}^{\frac{2}{q^{-}-1}}{\left({\lambda }_2{b}_1+{\mu }_2{d}_1\right)}^{\frac{2}{q^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

It is easy to see that ${\varphi }_k,{\varphi }_1\in C^1\left(\overline{\mathsf{\Omega }}\right).$ Denote

$$\begin{array}{ccc}\hfill \alpha & =& min\left\{\frac{infp\left(x\right)-1}{4\left(sup\left|\nabla p\left(x\right)\right|+1\right)},1\right\},\hfill \\ \hfill \zeta & =& min\left\{{\lambda }_{1}f\left(0\right)+{\mu }_{1}h\left(0\right),{\lambda }_{2}g\left(0\right)+{\mu }_2\tau \left(0\right),-1\right\}.\hfill \end{array}$$

By some simple computations, we can obtain

$$-{\Delta }_{p\left(x\right)}{\varphi }_k=\left\{\begin{array}{c}-k^{\prime }{\left(e^{k^{\prime }d\left(x\right)}\right)}^{p\left(x\right)-1}\left[\left(p\left(x\right)-1\right)+\left(d\left(x\right)+\frac{ln{k}^{\prime }}{k^{\prime }}\right)\nabla p\nabla d+\frac{\Delta d}{k^{\prime }}\right],d\left(x\right)<\sigma ,\hfill \\ \left\{\frac{1}{2\delta -\sigma }\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\left[\left(ln{k}^{\prime }{e}^{k^{\prime }\sigma }\right){\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}\nabla p\nabla d+\Delta d\right]\right\}\hfill \\ \times {\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-1}\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right),\sigma \le d\left(x\right)<2\delta ,\hfill \\ 0,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

$$-{\Delta }_{p\left(x\right)}{\varphi }_1=\left\{\begin{array}{c}-k{\left(e^{kd\left(x\right)}\right)}^{p\left(x\right)-1}\left[\left(p\left(x\right)-1\right)+\left(d\left(x\right)+\frac{lnk}{k}\right)\nabla p\nabla d+\frac{\Delta d}{k}\right],d\left(x\right)<\sigma ,\hfill \\ \left\{\frac{1}{2\delta -\sigma }\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\left[\left(lnk{e}^{k\sigma }\right){\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}\nabla p\nabla d+\Delta d\right]\right\}\hfill \\ \times {\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-1}\left({\lambda }_2{b}_1+{\mu }_2{d}_1\right),\sigma \le d\left(x\right)<2\delta ,\hfill \\ 0,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

From (H4), there exists a positive constant $L>1$ such that

$$f\left(L-1\right)\ge 1,g\left(L-1\right)\ge 1,h\left(L-1\right)\ge 1,\tau \left(L-1\right)\ge 1.$$

Let $\sigma =\frac{1}{k^{\prime }}lnL.$ Then,

$$\sigma k^{\prime }=lnL.$$

(6)

If $k^{\prime }$ is sufficiently large, from the problem defined in (6), we have

$$-{\Delta }_{p\left(x\right)}{\varphi }_1\le -k^{\prime p\left(x\right)}\alpha ,d\left(x\right)<\sigma .$$

(7)

Let $\frac{\lambda \zeta }{m_{\infty }}=k^{\prime }\alpha .$ Then,

$$-k^{\prime p\left(x\right)}\alpha \ge -{\lambda }^{p\left(x\right)}\frac{\zeta }{m_{\infty }}.$$

From the problem defined in (7), we have

$$\left\{\begin{array}{c}-M\left(I_0\left({\varphi }_k\right)\right){\Delta }_{p\left(x\right)}{\varphi }_k\le M\left(I_0\left({\varphi }_k\right)\right){\lambda }^{p\left(x\right)}\frac{\zeta }{m_{\infty }},\hfill \\ \le {\lambda }^{p\left(x\right)}\zeta ,\hfill \\ \le {\lambda }^{p\left(x\right)}\left({\lambda }_1{a}_{1}f\left(0\right)+{\mu }_1{c}_{1}h\left(0\right)\right),\hfill \\ \le {\lambda }^{p\left(x\right)}\left({\lambda }_{1}a\left(x\right)f\left({\varphi }_1\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_k\right)\right)-{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}},d\left(x\right)<\sigma .\hfill \end{array}\right.$$

Since $d\left(x\right)\in C^2\left(\overline{\partial {\mathsf{\Omega }}_{3\delta }}\right)$, there exists a positive constant $C_3$ such that

$$\begin{array}{ccc}\hfill -M\left(I_0\left({\varphi }_k\right)\right){\Delta }_{p\left(x\right)}{\varphi }_k& \le & m_{\infty }{\left(K{e}^{k^{\prime }\sigma }\right)}^{p\left(x\right)-1}{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-1}\left({\lambda }_1+{\mu }_1\right)\hfill \\ & & \times \left|\left\{\frac{1}{2\delta -\sigma }\frac{2\left(p\left(x\right)-1\right)}{p^{-}-1}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\left[\left(ln{k}^{\prime }{e}^{k^{\prime }\sigma }\right){\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}\nabla p\nabla d+\Delta d\right]\right\}\right|\hfill \\ & \le & C_3{m}_{\infty }{\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)ln{k}^{\prime },\sigma \le d\left(x\right)<2\delta .\hfill \end{array}$$

If $k^{\prime }$ is sufficiently large, let $\frac{\lambda \zeta }{m_{\infty }}=k^{\prime }\alpha$. Then, we have

$$\begin{array}{ccc}\hfill C_3{m}_{\infty }{\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)ln{k}^{\prime }& =& C_3{m}_{\infty }{\left(KL\right)}^{p\left(x\right)-1}\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)ln{k}^{\prime }\hfill \\ & \le & {\lambda }^{p\left(x\right)}\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)-{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}}.\hfill \end{array}$$

Then,

$$-M\left(I_0\left({\varphi }_k\right)\right){\Delta }_{p\left(x\right)}{\varphi }_k\le {\lambda }^{p\left(x\right)}\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)-{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}},\sigma \le d\left(x\right)<2\delta .$$

(8)

Since ${\varphi }_1\left(x\right),{\varphi }_2\left(x\right)$ and $f,h$ are monotone, when $\lambda$ is large enough, we have

$$\begin{array}{ccc}\hfill -M\left(\underset{\mathsf{\Omega }}{\int }\frac{1}{p\left(x\right)}{\left|\nabla {\varphi }_k\right|}^{p\left(x\right)}dx\right){\Delta }_{p\left(x\right)}{\varphi }_k& \le & {\lambda }^{p\left(x\right)}\left({\lambda }_{1}a\left(x\right)f\left({\varphi }_1\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_k\right)\right)\hfill \\ \hfill -{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}},\sigma & \le & d\left(x\right)<2\delta ,\hfill \end{array}$$

$$\begin{array}{c}-M\left(I_0\left({\varphi }_k\right)\right){\Delta }_{p\left(x\right)}{\varphi }_k=0\le {\lambda }^{p\left(x\right)}\left({\lambda }_1{a}_1+{\mu }_1{c}_1\right)\hfill \\ \le {\lambda }^{p\left(x\right)}\left({\lambda }_{1}a\left(x\right)f\left({\varphi }_1\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_k\right)\right)\hfill \\ -{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}},2\delta \le d\left(x\right).\hfill \end{array}$$

(9)

Combining two problems which defined in (8) and (9), we can conclude that

$$\begin{array}{c}-M\left(I_0\left({\varphi }_k\right)\right){\Delta }_{p\left(x\right)}{\varphi }_k\le {\lambda }^{p\left(x\right)}\left({\lambda }_{1}a\left(x\right)f\left({\varphi }_1\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_k\right)\right)\hfill \\ -{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}},\mathrm{a}.\mathrm{e}.\mathrm{on}\mathsf{\Omega }.\hfill \end{array}$$

(10)

Similarly,

$$\begin{array}{c}-M\left(I_0\left({\varphi }_1\right)\right){\Delta }_{p\left(x\right)}{\varphi }_1\le {\lambda }^{p\left(x\right)}\left({\lambda }_{2}b\left(x\right)g\left({\varphi }_k\right)+{\mu }_{2}d\left(x\right)\tau \left({\varphi }_1\right)\right)\hfill \\ -{\displaystyle \frac{{\varphi }_k-{\varphi }_{k-1}}{{\tau }^{\prime }}},\mathrm{a}.\mathrm{e}.\mathrm{on}\mathsf{\Omega }.\hfill \end{array}$$

(11)

From the problems defined in (10) and (11), we can see that $\left({\varphi }_k,{\varphi }_1\right)$ is a subsolution of problem (3).

Step 2. We will construct a supersolution of problem (3).

We consider

$$\left\{\begin{array}{c}-M\left(I_0\left(z_k\right)\right){\Delta }_{p\left(x\right)}{z}_k=\frac{{\lambda }^{p^{+}}}{m_0}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu -{\displaystyle \frac{z_k-z_{k-1}}{{\tau }^{\prime }}}\mathrm{in}\mathsf{\Omega },\hfill \\ -M\left(I_0\left(z_1\right)\right){\Delta }_{p\left(x\right)}{z}_1=\frac{{\lambda }^{p^{+}}}{m_0}\left({\lambda }_2{b}_2+{\mu }_2{d}_2\right)g\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right)\hfill \\ -{\displaystyle \frac{z_k-z_{k-1}}{{\tau }^{\prime }}}\mathrm{in}\mathsf{\Omega },\hfill \\ z_k=z_1=0\mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

where $\beta =\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)={max}_{x\in \overline{\mathsf{\Omega }}}{z}_k\left(x\right).$ We shall prove that $\left(z_k,z_1\right)$ is a supersolution of problem (3).

For $q\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ with $q\ge 0$, it is easy to see that

$$\begin{array}{c}M\left(I_0\left(z_1\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_1\right|}^{p\left(x\right)-2}\nabla z_1.\nabla qdx\hfill \\ =\frac{1}{m_0}M\left(I_0\left(z_1\right)\right)\underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}\left({\lambda }_2{b}_2+{\mu }_2{d}_2\right)g\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right)qdx\hfill \\ \ge \underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}{\lambda }_{2}b\left(x\right)g\left(z_k\right)qdx+\underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}{\mu }_{2}d\left(x\right)g\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1+{\mu }_1\right)\mu \right)\right)qdx.\hfill \end{array}$$

(12)

By (H6), for large enough $\mu$, using Lemma 2, we have

$$\begin{array}{c}g\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right)\ge \tau \left(C_2{\left[{\lambda }^{p^{+}}\left({\lambda }_2{b}_2+{\mu }_2{d}_2\right)g\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right)\right]}^{\frac{1}{p^{-}-1}}\right)\\ \ge \tau \left(z_1\right).\end{array}$$

(13)

Hence,

$$\begin{array}{c}M\left(I_0\left(z_1\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_1\right|}^{p\left(x\right)-2}\nabla z_1·\nabla qdx\ge \underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}{\lambda }_{2}b\left(x\right)g\left(z_k\right)qdx\\ +\underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}{\mu }_{2}d\left(x\right)\tau \left(z_1\right)qdx-\underset{\mathsf{\Omega }}{\int }{\displaystyle \frac{z_k-z_{k-1}}{{\tau }^{\prime }}}qdx.\end{array}$$

(14)

In addition,

$$\begin{array}{ccc}\hfill M\left(I_0\left(z_k\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_k\right|}^{p\left(x\right)-2}\nabla z_k·\nabla qdx& =& \frac{1}{m_0}M\left(I_0\left(z_k\right)\right)\underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu qdx\hfill \\ & \ge & \underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu qdx.\hfill \end{array}$$

By (H4), (H5) and Lemma 2, when $\mu$ is sufficiently large, we have

$$\begin{array}{ccc}\hfill \left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu & \ge & \frac{1}{{\lambda }^{p^{+}}}{\left[\frac{1}{C_2}\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right]}^{p^{-}-1}\hfill \\ & \ge & {\mu }_{1}h\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right)\hfill \\ & & +{\lambda }_{1}f\left(C_2{\left[{\lambda }^{p^{+}}\left({\lambda }_2{b}_2+{\mu }_2{d}_2\right)g\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right)\right]}^{\frac{1}{p^{-}-1}}\right).\hfill \end{array}$$

Then,

$$\begin{array}{c}M\left(I_0\left(z_k\right)\right)\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_k\right|}^{p\left(x\right)-2}\nabla z_k·\nabla qdx\ge \underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}{\lambda }_{1}a\left(x\right)f\left(z_1\right)qdx\\ +\underset{\mathsf{\Omega }}{\int }{\lambda }^{p^{+}}{\mu }_{1}c\left(x\right)h\left(z_k\right)qdx-\underset{\mathsf{\Omega }}{\int }{\displaystyle \frac{z_k-z_{k-1}}{{\tau }^{\prime }}}qdx.\end{array}$$

(15)

According to the problems (14) and (15), we can conclude that $\left(z_k,z_1\right)$ is a supersolution of problem (3). It only remains to prove that ${\varphi }_k\le z_k$ and ${\varphi }_1\le z_1.$

In the definition of $v_1\left(x\right)$, let

$$\gamma =\frac{2}{\delta }\left(\underset{\overline{\mathsf{\Omega }}}{max}{\varphi }_k\left(x\right)+\underset{\overline{\mathsf{\Omega }}}{max}\left|\nabla {\varphi }_k\right|\left(x\right)\right).$$

We claim that

$${\varphi }_k\left(x\right)\le v_1\left(x\right),\forall x\in \mathsf{\Omega }.$$

(16)

From the definition of $v_1$, it is easy to see that

$${\varphi }_k\left(x\right)\le 2\underset{\overline{\mathsf{\Omega }}}{max}{\varphi }_k\left(x\right)\le v_1\left(x\right),\mathrm{when}d\left(x\right)=\delta$$

and

$${\varphi }_k\left(x\right)\le 2\underset{\overline{\mathsf{\Omega }}}{max}{\varphi }_k\left(x\right)\le v_1\left(x\right),\mathrm{when}d\left(x\right)\ge \delta .$$

$${\varphi }_k\left(x\right)\le v_1\left(x\right),\mathrm{when}d\left(x\right)<\delta .$$

Since $v_1-{\varphi }_k\in C^1\left(\overline{\partial {\mathsf{\Omega }}_{\delta }}\right),$ there exists a point $x_0\in \overline{\partial {\mathsf{\Omega }}_{\delta }}$ such that

$$v_1\left(x_0\right)-{\varphi }_k\left(x_0\right)=\underset{x_0\in \overline{\partial {\mathsf{\Omega }}_{\delta }}}{min}\left(v_1\left(x_0\right)-{\varphi }_k\left(x_0\right)\right).$$

If $v_1\left(x_0\right)-{\varphi }_k\left(x_0\right)<0,$ it is easy to see that $0<$ $d\left(x\right)<\delta$ and then

$$\nabla v_1\left(x_0\right)-\nabla {\varphi }_k\left(x_0\right)=0.$$

From the definition of $v_1$, we have

$$\left|\nabla v_1\left(x_0\right)\right|=\gamma =\frac{2}{\delta }\left(\underset{\overline{\mathsf{\Omega }}}{max}{\varphi }_k\left(x_0\right)+\underset{\overline{\mathsf{\Omega }}}{max}\left|\nabla {\varphi }_k\right|\left(x_0\right)\right)>\left|\nabla {\varphi }_k\right|\left(x_0\right).$$

It is a contradiction to

$$\nabla v_1\left(x_0\right)-\nabla {\varphi }_k\left(x_0\right)=0.$$

Thus, problem (16) is valid.

Obviously, there exists a positive constant $C_3$ such that

$$\gamma \le C_3\lambda .$$

Since $d\left(x\right)\in C^2\left(\overline{\partial {\mathsf{\Omega }}_{3\delta }}\right)$, according to the proof of Lemma 2, there exists a positive constant $C_4$ such that

$$M\left(I_0\left(v_1\right)\right)-{\Delta }_{p\left(x\right)}{v}_1\left(x\right)\le C_{*}{\gamma }^{p\left(x\right)-1+\theta }\le C_4{\lambda }^{p\left(x\right)-1+\theta }.a.e\mathrm{in}\mathsf{\Omega },\mathrm{where}\theta \in \left(0,1\right).$$

When $\eta \ge {\lambda }^{p^{+}}$ is large enough, we have

$$-{\Delta }_{p\left(x\right)}{v}_1\left(x\right)\le \eta .$$

According to the comparison principle, we have

$$v_1\left(x\right)\le \omega \left(x\right),\forall x\in \mathsf{\Omega }.$$

(17)

From problems (16) and (17), when $\eta \ge {\lambda }^{p^{+}}$ and $\lambda \ge 1$ is sufficiently large, we have

$${\varphi }_k\left(x\right)\le v_1\left(x\right)\le \omega \left(x\right),\forall x\in \mathsf{\Omega }.$$

(18)

According to the comparison principle, when $\mu$ is large enough, we have

$$v_1\left(x\right)\le \omega \left(x\right)\le z_k\left(x\right),\forall x\in \mathsf{\Omega }.$$

Combining the definition of $v_1\left(x\right)$ and the problem defined in (18), it is easy to see that

$${\varphi }_k\left(x\right)\le v_1\left(x\right)\le \omega \left(x\right)\le z_k\left(x\right),\forall x\in \mathsf{\Omega }.$$

When $\mu \ge 1$ and $\lambda$ is large enough, from Lemma 2.6 (see [22]), we can see that $\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)$ is large enough, and then

$$\frac{{\lambda }^{p^{+}}}{m_0}\left({\lambda }_2{b}_2+{\mu }_2{d}_2\right)g\left(\beta \left({\lambda }^{p^{+}}\left({\lambda }_1{a}_2+{\mu }_1{c}_2\right)\mu \right)\right)$$

is large enough. Similarly, we have ${\varphi }_1\le z_1$. This completes the proof. □