(p(x),q(x))-Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

Abstract

In the present article, we study a class of Kirchhoff-type equations driven by the $\left(p\right(x),q(x\left)\right)$-Laplacian. Due to the lack of a variational structure, ellipticity, and monotonicity, the well-known variational methods are not applicable. With the help of the Galerkin method and Brezis theorem, we obtain the existence of finite-dimensional approximate solutions and weak solutions. One of the main difficulties and innovations of the present article is that we consider competing $\left(p\right(x),q(x\left)\right)$-Laplacian, convective terms, and logarithmic nonlinearity with variable exponents, another one is the weaker assumptions on nonlocal term $M_{\upsilon \left(x\right)}$ and nonlinear term g.

1. Introduction

The purpose of the present article is to investigate the following $\left(p\right(x),q(x\left)\right)$-Kirchhoff-type equations involving logarithmic nonlinearity and convection terms:

$$\left\{\begin{array}{c}-M_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\Delta }_{p\left(x\right)}\eta -\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\Delta }_{q\left(x\right)}\eta \hfill \\ ={\lambda \left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|+g(x,\eta ,\nabla \eta ),\mathrm{in}\Omega ,\hfill \\ {\eta |}_{\partial \Omega }=0,\hfill \end{array}\right.$$

(1)

where $r\left(x\right)\in C_{+}(\Omega )$, $\mu ,\lambda$ are real parameters, and $\Omega$ is an open bounded domain in ${\mathbb{R}}^N$ with a smooth boundary.

Here, ${\Delta }_{\gamma \left(x\right)}$ is a $\gamma \left(x\right)$-Laplace operator, defined by

$$\begin{array}{c}\hfill {\Delta }_{\gamma \left(x\right)}\eta ={div\left(\right|\nabla \eta |}^{\gamma \left(x\right)-2}\nabla \eta )=\sum _{i=1}^N\left({|\nabla \eta |}^{\gamma \left(x\right)-2}\frac{\partial \eta }{\partial x_i}\right),\gamma \left(x\right)\in \left\{p\left(x\right),q\left(x\right)\right\},\end{array}$$

(2)

for all $x\in \Omega$ and $\eta \in C_0^{\infty }\left({\mathbb{R}}^N\right)$, and denote

$$\begin{array}{c}\hfill {\delta }_{s\left(x\right)}\left(\eta \right)={\int }_{\Omega }\frac{1}{s\left(x\right)}{|\nabla \eta |}^{s\left(x\right)}dx,s\left(x\right)\in \left\{p\left(x\right),q\left(x\right)\right\}.\end{array}$$

(3)

From now on, we briefly state some major features of problem (1). One of the significant characteristics of the problem (1) is the presence of double non-local Kirchhoff terms, which were introduced in [1] as follows:

$$\begin{array}{c}\hfill \rho \frac{{\partial }^2\eta \left(x\right)}{\partial t^2}-\left(\frac{p_0}{h}+\frac{E}{2L}{\int }_0^L{\left|\frac{\partial \eta \left(x\right)}{\partial t}\right|}^{2}dx\right)\frac{{\partial }^2\eta \left(x\right)}{\partial x^2}=0,\end{array}$$

(4)

where parameters $\rho ,p_0,h,E,$ and L are real positive constants. Equation (4) is a nonlocal problem, which contains a nonlocal coefficient $\frac{p_0}{h}+\frac{E}{2L}{\int }_0^L{\left|\frac{\partial \eta \left(x\right)}{\partial t}\right|}^{2}dx$, and has a wide range of applications and research in physical systems, such as non-homogeneous Kirchhoff-type equations in ${\mathbb{R}}^N$ [2], nonlocal Kirchhoff equations of elliptic type [3], Kirchhoff–Schrödinger type equations [4], $p\left(x\right)$-Laplacian Dirichlet problem [5,6], Kirchhoff–Choquard equations involving variable-order [7,8], fractional $p(·)$-Kirchhoff type problem in ${\mathbb{R}}^N$ [9], Kirchhoff-type equations involving the fractional $p_1\left(x\right)\&p_2\left(x\right)$-Laplace operator [10], fractional $p(x,·)$-Kirchhoff-type problems in ${\mathbb{R}}^N$ [11], and fractional Sobolev space and applications to nonlocal variational problems [12]. For more Kirchhoff-type problems, we also mention that [13] studied a class of Kirchhoff nonlocal fractional equations and obtained the existence of three solutions, Ref. [14] discussed a class of p-Kirchhoff equations via the fountain theorem and dual fountain theorem, and Ref. [15] researched the existence of non-negative solutions for a Kirchhoff type problem driven by a non-local integro-differential operator.

Let $M_i:{\mathbb{R}}_0^{+}\to {\mathbb{R}}^{+}$ and $p\left(x\right),q\left(x\right):{\mathbb{R}}^N\to (1,+\infty )$ be continuous functions, which satisfy the following conditions:

$H_m$:

There are some constants $m_{\upsilon \left(x\right)}=m_{\upsilon \left(x\right)}\left(\iota \right)>0$$\left(\upsilon \right(x)\in \{p\left(x\right),q\left(x\right)\left\}\right)$ for all $\iota >0$ such that

$$M_{\upsilon \left(x\right)}\left(t\right)\ge m_{\upsilon \left(x\right)},\mathrm{for}\mathrm{any}t>\iota .$$

$H_{pq}$:

The conditions that we impose on $p\left(x\right),q\left(x\right)$ are as follows:

$$1<p^{-}:=\underset{x\in \overline{\Omega }}{inf}p\left(x\right)\le p^{+}:=\underset{x\in \overline{\Omega }}{sup}p\left(x\right)<+\infty ,$$

$$1<q^{-}:=\underset{x\in \overline{\Omega }}{inf}q\left(x\right)\le q^{+}:=\underset{x\in \overline{\Omega }}{sup}q\left(x\right)<+\infty .$$

Another significant characteristic of the problem (1) is the presence of double operators, which comes from the following system

$$\begin{array}{c}\hfill {\eta }_t=\mathrm{div}[D\eta \nabla \eta ]+c(x,\eta ),\end{array}$$

(5)

where $D\eta ={|\nabla \eta |}^{p-2}+{|\nabla \eta |}^{q-2}$ and $c(x,\eta )$ is a polynomial of $\eta$. System (5) had a wide range of applications in the field of physics and related sciences, for example, on the stationary solutions of generalized reaction diffusion equations [16], elliptic problems with critical growth in ${\mathbb{R}}^N$ [17], nontrivial solutions to nonlinear elliptic equation in ${\mathbb{R}}^N$ [18], and fractional Choquard problems with variable order [19]. The function $\eta$ in (5) describes a concentration, and the first term corresponds to the diffusion with a (generally non-constant) diffusion coefficient $D\eta$, whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological applications, the reaction term $c(x,\eta )$ in (5) has a polynomial form with respect to the concentration $\eta$.

When $M_{\upsilon \left(x\right)}=1(\upsilon \left(x\right)\in \{p\left(x\right),q\left(x\right)\})$ and $\mu =1$, Chung et al. in [20] devoted to the study of equations involving both $p_1\left(x\right)$-Laplacian and $p_2\left(x\right)$-Laplacian

$$\left\{\begin{array}{c}{(-\Delta )}_{p_1(·)}^s\eta \left(x\right)+{(-\Delta )}_{p_2(·)}^s\eta \left(x\right)+{\left|\eta \left(x\right)\right|}^{q\left(x\right)-2}\eta \left(x\right)\hfill \\ =\lambda V_1{\left(x\right)\left|\eta \left(x\right)\right|}^{r_1\left(x\right)-2}\eta \left(x\right)-\lambda V_2\left(x\right){\left|\eta \left(x\right)\right|}^{r_2\left(x\right)-2}\eta \left(x\right),x\in \Omega ,\hfill \\ \eta \left(x\right)=0,x\in \partial \Omega ,\hfill \end{array}\right.$$

(6)

where $p_1,p_2,q,r_1$, and $r_2$ are different continuous functions, while $V_1,V_2$ are suitable weights. Equation (6) considered the local double Laplace operators, whose results differed from those of the single Laplace operator.

When $M_{\upsilon \left(x\right)}=1(\upsilon \left(x\right)\in \{p\left(x\right),q\left(x\right)\})$ and $\mu =-1$, we mention that Motreanu in [21] considered Dirichlet problems with competing operators

$$\left\{\begin{array}{c}-{\Delta }_p\eta \left(x\right)+{\Delta }_q\eta \left(x\right)=g(x,\eta \left(x\right),\nabla \eta \left(x\right)),\mathrm{in}\Omega ,\hfill \\ \eta \left(x\right)=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(7)

where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain. Equation (7) includes the sum $-{\Delta }_p+{\Delta }_q$ of the negative p-Laplacian ${\Delta }_p$ and of the q-Laplacian ${\Delta }_q$, due to competition between $-{\Delta }_p$ and ${\Delta }_q$, and the operator $-{\Delta }_p+{\Delta }_q$ has a different behavior in comparison to the operator $-{\Delta }_p+{\Delta }_q$. Moreover, the ellipticity and monotonicity property of the operator $-{\Delta }_p+{\Delta }_q$ are lost.

The third significant characteristic of the problem (1) is the presence of convection term $g(x,\eta ,\nabla \eta )$, depending on the function $\eta$ and on its gradient $\nabla \eta$, which makes the problem (1) non-variational, plays an important role in science and technology fields, and is widely used to describe physical phenomena. For example, due to convection and diffusion processes, particles or energy are converted and transferred inside physical systems. For the work related to this topic, we cite the interesting work [21,22,23,24] and their references.

The work in [25] focused on the p-Kirchhoff-type equations with gradient dependence in the reaction that is

$$\left\{\begin{array}{c}-M\left({\int }_{\Omega }{|\nabla \eta \left(x\right)|}^{p}dx\right){\Delta }_p\eta \left(x\right)=g(x,\eta \left(x\right),\nabla \eta \left(x\right)),\mathrm{in}\Omega ,\hfill \\ \eta \left(x\right)=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(8)

where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with a smooth boundary. The existence of solutions for the problem (8) was obtained by utilizing Galerkin’s approach.

One more reference on convection is Vetro [26], which was devoted to the study of the following $p\left(x\right)$-Kirchhoff-type equation:

$$\begin{array}{c}\hfill -{\Delta }_{p\left(x\right)}^K{\eta \left(x\right)=g(x,\eta \left(x\right),\nabla \eta \left(x\right)),\mathrm{in}\Omega ,\eta |}_{\partial \Omega }=0.\end{array}$$

(9)

The existence of weak solutions and generalized solutions for the problem (9) with gradient dependence was obtained via applying a topological method.

The nonlinearity $g:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ is a Carathéodory function, satisfying

$H_{g_1}$:

There exist some constants $c<1,d>0$ and a function $\alpha \in [1,p^{-})$ such that

$$g(x,\varpi ,\nu )\varpi \le {c\left|\nu \right|}^{p\left(x\right)}+{d\left(\right|\varpi |}^{\alpha \left(x\right)}+1),\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \Omega \mathrm{and}\mathrm{all}(\varpi ,\nu )\in \mathbb{R}\times {\mathbb{R}}^N.$$

$H_{g_2}$:

There exists a positive function $\varphi \left(x\right)\in L^{p^{\prime }\left(x\right)}(\Omega )$ and some positive constants a and b such that

$$\left|g(x,\varpi ,\nu )\right|\le h\left(x\right)+{a\left|\varpi \right|}^{\varphi \left(x\right)}+b{\left|\nu \right|}^{\frac{\psi \left(x\right)}{p^{\prime }\left(x\right)}},\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \Omega \mathrm{and}\mathrm{all}(\varpi ,\nu )\in \mathbb{R}\times {\mathbb{R}}^N.$$

where $\varphi \left(x\right)\in C\left(\overline{\Omega }\right),\psi \left(x\right)\in C\left(\overline{\Omega }\right)$ such that $0<{\varphi }^{-}\le {\varphi }^{+}<p^{-}-1$, ${\left(\frac{\psi }{p^{\prime }}\right)}^{+}<p^{-}-1.$

The last significant characteristic of the problem (1) is the presence of logarithmic nonlinearity. The interest in studying problems with logarithmic nonlinearity is motivated not only by the purpose of describing mathematical and physical phenomena but also by their application in realistic models. For instance, in the biological population, we use the function $\eta \left(x\right)$ to represent the density of the population, and the logarithmic nonlinear term ${\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|$ to denote external influencing factors.

Many scholars make efforts to investigate logarithmic nonlinearity, and, indeed, some important results were obtained; for example, see [27,28,29,30]. Peculiarly, Xiang et al. in [31] considered the following equation:

$$\left\{\begin{array}{c}M\left({\left[\eta \right]}_{s,p}^p\right){(-\Delta )}_p^s\eta ={h\left(x\right)\left|\eta \right|}^{\theta p-2}\eta ln\left|\eta \right|+{\lambda \left|\eta \right|}^{q-2}\eta ,x\in \Omega ,\hfill \\ \eta \left(x\right)=0,x\in {\mathbb{R}}^N\backslash \Omega ,\hfill \end{array}\right.$$

(10)

where $M\left({\left[\eta \right]}_{s,p}^p\right)={\left[\eta \right]}_{s,p}^{p(\theta -1)}$ and $h\left(x\right)$ is a sign-changing function. The existence of least energy solutions (10) was obtained by utilizing the Nehari manifold method.

Until now, there have been few papers to handle the equations involving logarithmic nonlinearity with variable exponents. Recently, Boudjeriou in [32] studied the following initial value problem:

$$\left\{\begin{array}{c}{\eta }_t\left(x\right)-{\Delta }_{p\left(x\right)}\eta \left(x\right)={\left|\eta \left(x\right)\right|}^{s\left(x\right)-2}\eta \left(x\right)log\left(\right|\eta \left(x\right)\left|\right),\mathrm{in}\Omega ,t>0,\hfill \\ \eta \left(x\right)=0,\mathrm{in}\partial \Omega ,t>0,\hfill \\ \eta (x,0)={\eta }_0\left(x\right),\mathrm{in}\Omega .\hfill \end{array}\right.$$

(11)

The weak solutions of Equation (11) were obtained under suitable conditions. Moreover, Zeng et al. in [33] were devoted to the study of equations with logarithmic nonlinearity and variable exponents by applying the logarithmic inequality.

Motivated by the previous and aforementioned cited works, there is no result for the Kirchhoff-type equations, which combine with variable exponents, competing $\left(p\right(x),q(x\left)\right)$-Laplacian, logarithmic nonlinearity, and convection terms; therefore, we will investigate the existence of solutions for these kinds of equations, which are different from the work of [25,26,31,32]. Under weaker conditions on the nonlocal term $M_{\upsilon \left(x\right)}$ and the nonlinearities g, we prove the existence of finite-dimensional approximate solutions by using the Galerkin method and obtain the existence of weak solutions with the help of the Brezis theorem. One of the main difficulties and innovations of the present article is that we consider competing $\left(p\right(x),q(x\left)\right)$-Laplacian, convective term, and logarithmic nonlinearity with variable exponents; another one is the weaker assumptions on nonlocal term $M_{\upsilon \left(x\right)}$ and nonlinear term g.

The present article is divided into six sections. Aside from Section 1, we have Section 2 given some preliminary notions and results about Lebesgue spaces and Sobolev spaces, and proved some technical lemmas. The finite-dimensional approximate solutions are obtained in Section 3. Section 4 discusses the existence of weak solutions by applying the Brezis theorem, and we give two examples of application of our theorems in Section 5 and present conclusions in Section 6.

2. Preliminary Results and Some Technical Lemmas

In this section, we briefly review some basic knowledge of generalized Lebesgue spaces and Sobolev spaces with variable exponents, and then give two technical lemmas.

For any real-valued function H defined on a domain $\Omega$, we denote

$$C_{+}\left(\overline{\Omega }\right):=\left\{H\left(x\right)\in C(\overline{\Omega },\mathbb{R}):1<H^{-}:=\underset{x\in \overline{\Omega }}{inf}H\left(x\right)\le H\left(x\right)\le H^{+}:=\underset{x\in \overline{\Omega }}{sup}H\left(x\right)<+\infty \right\}.$$

Letting $\vartheta \left(x\right)\in C_{+}\left(\overline{\Omega }\right)$, we define the generalized Lebesgue spaces with variable exponents as

$$L^{\vartheta \left(x\right)}(\Omega ):=\left\{\eta :\eta \mathrm{is}\mathrm{a}\mathrm{measurable}\mathrm{function}\mathrm{and}{\int }_{\Omega }{\left|\eta \right|}^{\vartheta \left(x\right)}dx<\infty \right\},$$

provided with the Luxemburg norm

$${\parallel \eta \parallel }_{\vartheta \left(x\right)}={\parallel \eta \parallel }_{L^{\vartheta \left(x\right)}(\Omega )}:=\inf \left\{\chi >0:{\int }_{\Omega }{\left|\frac{\eta }{\chi }\right|}^{\vartheta \left(x\right)}dx\le 1\right\};$$

then, $(L^{\vartheta \left(x\right)}(\Omega ),\parallel ·{\parallel }_{\vartheta \left(x\right)})$ is a separable and reflexive Banach spaces; see [34,35].

Lemma 1

(see [35]). Let $\vartheta \left(x\right)$ be the conjugate exponent of $\tilde{\vartheta }\left(x\right)\in C_{+}\left(\overline{\Omega }\right)$, that is,

$$\frac{1}{\vartheta \left(x\right)}+\frac{1}{\tilde{\vartheta }\left(x\right)}=1,forallx\in \Omega .$$

Assume that $\eta \in L^{\vartheta \left(x\right)}(\Omega )$ and $\xi \in L^{\tilde{\vartheta }\left(x\right)}(\Omega )$; then,

$$\left|{\int }_{\Omega }\eta \xi dx\right|\le \left(\frac{1}{{\vartheta }^{-}}+\frac{1}{{\tilde{\vartheta }}^{-}}\right){\parallel \eta \parallel }_{\vartheta \left(x\right)}{\parallel \xi \parallel }_{\tilde{\vartheta }\left(x\right)}\le {2\parallel \eta \parallel }_{\vartheta \left(x\right)}{\parallel \xi \parallel }_{\tilde{\vartheta }\left(x\right)}.$$

Proposition 1

(see [36]). The modular of $L^{\vartheta \left(x\right)}(\Omega )$, which is the mapping ${\rho }_{\vartheta \left(x\right)}:L^{\vartheta \left(x\right)}(\Omega )\to \mathbb{R}$, is defined by

$${\rho }_{\vartheta \left(x\right)}\left(\eta \right):={\int }_{\Omega }{\left|\eta \right|}^{\vartheta \left(x\right)}dx.$$

Assume that ${\eta }_n,\eta \in L^{\vartheta \left(x\right)}(\Omega )$; then, the following properties hold:

$$\begin{array}{cc}\hfill \left(1\right)& {\parallel \eta \parallel }_{\vartheta \left(x\right)}>1\Rightarrow {\parallel \eta \parallel }_{\vartheta \left(x\right)}^{{\vartheta }^{-}}\le {\rho }_{\vartheta \left(x\right)}\left(\eta \right)\le {\parallel \eta \parallel }_{\vartheta \left(x\right)}^{{\vartheta }^{+}},\hfill \\ \hfill \left(2\right)& {\parallel \eta \parallel }_{\vartheta \left(x\right)}<1\Rightarrow {\parallel \eta \parallel }_{\vartheta \left(x\right)}^{{\vartheta }^{+}}\le {\rho }_{\vartheta \left(x\right)}\left(\eta \right)\le {\parallel \eta \parallel }_{\vartheta \left(x\right)}^{{\vartheta }^{-}},\hfill \\ \hfill \left(3\right)& {\parallel \eta \parallel }_{\vartheta \left(x\right)}<1(\mathrm{resp}.=1,>1)\iff {\rho }_{\vartheta \left(x\right)}\left(\eta \right)<1(\mathrm{resp}.=1,>1),\hfill \\ \hfill \left(4\right)& \parallel {\eta }_n{\parallel }_{\vartheta \left(x\right)}\to 0(\mathrm{resp}.\to +\infty )\iff {\rho }_{\vartheta \left(x\right)}\left({\eta }_n\right)\to 0(\mathrm{resp}.\to +\infty ),\hfill \\ \hfill \left(5\right)& \underset{n\to \infty }{lim}{|{\eta }_n-\eta |}_{\vartheta \left(x\right)}=0\iff \underset{n\to \infty }{lim}{\rho }_{\vartheta \left(x\right)}({\eta }_n-\eta )=0.\hfill \end{array}$$

Now, we consider the following generalized Sobolev spaces with variable exponents

$$W=W^{1,\vartheta \left(x\right)}(\Omega ):=\left\{\eta \in L^{\vartheta \left(x\right)}(\Omega ):|\nabla \eta |\in L^{\vartheta \left(x\right)}(\Omega )\right\},$$

endowed with the norm

$${\parallel \eta \parallel }_W:={\parallel \eta \parallel }_{\vartheta \left(x\right)}+{\parallel \nabla \eta \parallel }_{\vartheta \left(x\right)};$$

then, (${W,\parallel ·\parallel }_W$) is a separable and reflexive Banach spaces, see [34].

Lemma 2

(see [34]). Assume that $\gamma \left(x\right)\in C_{+}\left(\overline{\Omega }\right)$ fulfills

$$1<{\gamma }^{-}=\underset{x\in \overline{\Omega }}{min}\gamma \left(x\right)\le \gamma \left(x\right)<{\vartheta }^{*}\left(x\right)=\frac{N\vartheta \left(x\right)}{N-\vartheta \left(x\right)},foranyx\in \overline{\Omega }.$$

Then, there exists $C_{\gamma }=C_{\gamma }(N,\vartheta ,\gamma ,\Omega )>0$ such that

$${\parallel \eta \parallel }_{\gamma \left(x\right)}\le C_{\gamma }{\parallel \eta \parallel }_W,$$

for any $\eta \in W$. Moreover, the embedding $W\hookrightarrow L^{\gamma \left(x\right)}(\Omega )$ is compact.

Let $W_0$ denote the closure of $C_0^{\infty }(\Omega )$ in W with respect to the norm ${\parallel \eta \parallel }_{W_0}$, which is the subspace of W. Thus, the spaces ($W_0,{\parallel ·\parallel }_{W_0}$) are also separable and reflexive Banach spaces.

Remark 1.

According to the Poincaré inequality, we know that ${\parallel \nabla \eta \parallel }_{\vartheta \left(x\right)}$ and ${\parallel \eta \parallel }_{W_0}$ are equivalent norms in $W_0$. From now on, we work on $W_0$ and replace ${\parallel \eta \parallel }_{W_0}$ by ${\parallel \nabla \eta \parallel }_{\vartheta \left(x\right)}$, that is,

$${\parallel \eta \parallel }_{W_0}={\parallel \nabla \eta \parallel }_{\vartheta \left(x\right)},\mathit{for}\mathit{all}\eta \in W_0.$$

Remark 2.

To simplify the presentation, we will denote the norm of $W_0$ by $\parallel ·\parallel$ instead of ${\parallel ·\parallel }_{W_0}$. $W_0^{*}$ denotes the dual space of $W_0$.

Our technique of proof is based on Galerkin methods together with the fixed point theorem, whose proof may be found in Lions [37].

Lemma 3.

Let $W_0$ be a finite dimensional space with the norm $\parallel ·\parallel$ and let $G:W_0\to W_0^{*}$ be a continuous mapping. Assume that there is a constant $R>0$ such that

$$\langle G\left(\eta \right),\eta \rangle \ge 0,\mathit{for}\mathit{all}\eta \in W_0\mathit{with}\parallel \eta \parallel =R,$$

then $\eta \in W_0$ exists with $\parallel \eta \parallel \le R$ satisfying $G\left(\eta \right)=0.$

The following two Lemmas provide a useful growth estimate, related to logarithmic nonlinear terms, which play an important role during our proof process.

Lemma 4.

Assume that $h\left(x\right)\in C_{+}\left(\overline{\Omega }\right)$; then, we have the following estimate:

$$\begin{array}{c}\hfill lnt\le \frac{1}{eh\left(x\right)}{t}^{h\left(x\right)}\le \frac{1}{e{h}^{-}}{t}^{h\left(x\right)},\mathit{for}\mathit{all}t\in [1,+\infty ).\end{array}$$

Proof. 

Let $h\left(x\right)\in C_{+}\left(\overline{\Omega }\right)$, and we construct the following function:

$$\begin{array}{c}\hfill f\left(t\right)=lnt-\frac{1}{eh\left(x\right)}{t}^{h\left(x\right)},\mathrm{for}\mathrm{all}t\in [1,+\infty ).\end{array}$$

With respect to t, just by taking a simple derivative, we deduce

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)=\frac{1}{t}-\frac{1}{e}{t}^{h\left(x\right)-1},\mathrm{for}\mathrm{all}t\in [1,+\infty ),\end{array}$$

and let $f^{\prime }\left(t\right)=0$; then, $t^{*}=e^{h^{-1}\left(x\right)}$. It is obvious that $t^{*}$ is the unique maximum point of the function $f\left(t\right)$, so $f\left(t\right)\le f\left(t^{*}\right)=0$ for all $t\in [1,+\infty )$. Therefore, based on the above discussion, we can obtain the stated conclusion. □

Lemma 5.

Assume that, for all $\eta \in W_0$ and $h\left(x\right),r\left(x\right)\in C_{+}\left(\overline{\Omega }\right)$, then the following inequality holds:

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx\le C_{{\Omega }_1}|\Omega |+\frac{1}{e{h}^{-}}max\left\{C_{h^{+}+r^{+}}{\parallel \eta \parallel }^{h^{+}+r^{+}},C_{h^{-}+r^{-}}{\parallel \eta \parallel }^{h^{-}+r^{-}}\right\},\end{array}$$

where $C_{{\Omega }_1},C_{h^{+}+r^{+}},C_{h^{-}+r^{-}}$ are some positive constants and $h\left(x\right)+r\left(x\right)\le h^{+}+r^{+}<2p^{-}<p^{*}\left(x\right)=\frac{Np\left(x\right)}{N-p\left(x\right)}.$

Proof. 

Let ${\Omega }_1=\{x\in \Omega :|\eta \left(x\right)|\le 1\}$ and ${\Omega }_2=\{x\in \Omega :|\eta \left(x\right)|\ge 1\}$; then,

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx={\int }_{{\Omega }_1}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx+{\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx.\end{array}$$

Since $\left|\eta \right(x\left)\right|\le 1$, there exist $M_{r_1}>0$ and $M_{r_2}>0$ such that ${\left|\eta \right|}^{r\left(x\right)}<M_{r_1}$ and $ln\left|\eta \right|<M_{r_2}$. By a simple calculation, we obtain

$$\begin{array}{c}\hfill {\int }_{{\Omega }_1}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx<C_{{\Omega }_1}|\Omega |,\end{array}$$

(12)

where $|\Omega |$ denotes the Lebesgue measure of $\Omega$ and $C_{{\Omega }_1}>0$. Using Lemma 4 with $h\left(x\right)+r\left(x\right)\le h^{+}+r^{+}<p^{*}\left(x\right)$, we deduce

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx\le & \frac{1}{e{h}^{-}}{\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)+h\left(x\right)}dx\hfill \\ \hfill \le & \frac{1}{e{h}^{-}}max\left\{{\parallel \eta \parallel }_{h\left(x\right)+r\left(x\right)}^{h^{+}+r^{+}},{\parallel \eta \parallel }_{h\left(x\right)+r\left(x\right)}^{h^{-}+r^{-}}\right\},\hfill \end{array}$$

in view of Lemma 2, and there exist some constants $C_{h^{+}+r^{+}}>0$ and $C_{h^{-}+r^{-}}>0$ such that

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx\le & \frac{1}{e{h}^{-}}max\left\{C_{h^{+}+r^{+}}{\parallel \eta \parallel }^{h^{+}+r^{+}},C_{h^{-}+p^{-}}{\parallel \eta \parallel }^{h^{-}+r^{-}}\right\}.\hfill \end{array}$$

(13)

It follows from (12) and (13) that

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx\le C_{{\Omega }_1}|\Omega |+\frac{1}{e{h}^{-}}max\left\{C_{h^{+}+r^{+}}{\parallel \eta \parallel }^{h^{+}+r^{+}},C_{h^{-}+r^{-}}{\parallel \eta \parallel }^{h^{-}+r^{-}}\right\}.\end{array}$$

This yields the stated conclusion. □

3. Finite Dimensional Approximate Solutions

Since $W_0$ is a reflexive and separable Banach space, see [34], and there exists an orthonormal basis $\{e_1,...,e_n,...\}$ in $W_0$, such that

$$\begin{array}{c}\hfill W_0=\overline{span\{e_1,...,e_n\}}.\end{array}$$

Define $X_n=span\{e_1,...,e_n\}$, which means a sequence of vector $X_n$ subspaces of $W_0$, satisfying

$$dim\left(X_n\right)<\infty \mathrm{for}\mathrm{all}n\ge 1,X_n\subset X_{n+1}\mathrm{for}\mathrm{all}n\ge 1,\mathrm{and}\overline{{\bigcup }_{n=1}^{\infty }{X}_n}=W_0.$$

It is known that $X_n$ and ${\mathbb{R}}^N$ are isomorphic and, for $\eta \in {\mathbb{R}}^N$, we have a unique $\xi \in X_n$ by the identification

$$\eta \to {\Sigma }_{i=1}^N{\xi }_i{e}_i=\xi ,\parallel \eta \parallel =\left|\xi \right|,$$

where $|·|$ is the Euclidian norm in ${\mathbb{R}}^N.$

Theorem 1.

Assume that conditions $H_m$, $H_{pq}$, and $H_{g_1}$ are satisfied; then,

• if $2p^{-}>p^{+}$ and $p^{-}>{\alpha }^{+}$, problem (1) admits a approximate solution for all $\mu \ge 0$ and $\lambda \le 0,$
• if $2p^{-}>p^{+}$, $p^{-}>q^{+}$ and $p^{-}>{\alpha }^{+}$, the problem (1) admits a approximate solution for all $\mu <0$ and $\lambda \le 0,$
• if $2q^{-}>p^{+}$ and $q^{-}>{\alpha }^{+}$, problem (1) admits a approximate solution for all $\mu \ge 0$ and $\lambda \le 0,$
• if $2p^{-}>r^{+}+h^{+}$ and $p^{-}>{\alpha }^{+}$, problem (1) admits a approximate solution for all $\mu \ge 0$ and $\lambda >0,$
• if $2p^{-}>r^{+}+h^{+}$, $p^{-}>q^{+}$ and $p^{-}>{\alpha }^{+}$, problem (1) admits a approximate solution for all $\mu <0$ and $\lambda >0,$
• if $2q^{-}>r^{+}+h^{+}$ and $q^{-}>{\alpha }^{+}$, problem (1) admits a approximate solution for all $\mu \ge 0$ and $\lambda >0,$

that is, for all $n\ge 1$ and $\phi \in X_n$, there exists ${\eta }_n\in X_n$ such that

$$\begin{array}{cc}\hfill M_{p\left(x\right)}& \left({\delta }_{p\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,\phi \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,\phi \rangle }_{q\left(x\right)}\hfill \\ \hfill & =\lambda {\int }_{\Omega }\left(|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right)\phi dx+{\int }_{\Omega }g(x,{\eta }_n,\nabla {\eta }_n)\phi dx.\hfill \end{array}$$

(14)

Proof. 

For all $\eta \in X_n$, we consider the mapping $G=(G_1,G_2,...,G_N):{\mathbb{R}}^N\to \mathbb{R}$ by

$$\begin{array}{cc}\hfill G_i=& M_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\langle \eta ,e_i\rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\langle \eta ,e_i\rangle }_{q\left(x\right)}\hfill \\ \hfill & -\lambda {\int }_{\Omega }\left({\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right)e_{i}dx-{\int }_{\Omega }g(x,\eta ,\nabla \eta )e_{i}dx.\hfill \end{array}$$

The following work shows that, for each $n\ge 1$, problem (1) has an approximate solution ${\eta }_n$ in $X_n$, namely

$$\begin{array}{c}{M}_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,e_i\rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,e_i\rangle }_{q\left(x\right)}\\ =\lambda {\int }_{\Omega }\left(|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right)e_{i}dx+{\int }_{\Omega }g(x,{\eta }_n,\nabla {\eta }_n)e_{i}dx.\end{array}$$

(15)

For $\eta \in X_n$, we have

$$\begin{array}{cc}\hfill \langle G,\eta \rangle =& M_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\langle \eta ,\eta \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\langle \eta ,\eta \rangle }_{q\left(x\right)}\hfill \\ \hfill & -{\int }_{\Omega }{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx-{\int }_{\Omega }g(x,\eta ,\nabla \eta )\eta dx,\hfill \\ \hfill \ge & \frac{1}{p^{+}}{\left({\int }_{\Omega }{|\nabla \eta |}^{p\left(x\right)}dx\right)}^2+\frac{\mu }{q^{+}}{\left({\int }_{\Omega }{|\nabla \eta |}^{p\left(x\right)}dx\right)}^2\hfill \\ \hfill & -\lambda {\int }_{\Omega }\left|{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|\right|dx-{\int }_{\Omega }\left|g(x,\eta ,\nabla \eta )\eta \right|dx.\hfill \end{array}$$

From $H_{g_1}$ and Lemma 5, we have the following estimate:

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{1}{p^{+}}{\left({\int }_{\Omega }{|\nabla \eta |}^{p\left(x\right)}dx\right)}^2+\frac{\mu }{q^{+}}{\left({\int }_{\Omega }{|\nabla \eta |}^{q\left(x\right)}dx\right)}^2\hfill \\ \hfill & -\frac{\lambda }{e{h}^{-}}max\left\{C_{h^{+}+r^{+}}{\parallel \eta \parallel }^{h^{+}+r^{+}},C_{h^{-}+r^{-}}{\parallel \eta \parallel }^{h^{-}+r^{-}}\right\}\hfill \\ \hfill & -\lambda C_{{\Omega }_1}|\Omega |-c{\int }_{\Omega }{|\nabla \eta |}^{p\left(x\right)}dx-d{\int }_{\Omega }{\left(\right|\eta |}^{\alpha \left(x\right)}+1)dx.\hfill \end{array}$$

According to Remark 1 and Lemma 2, there exist some positive constants $C_{{\alpha }^{+}}$ and $C_{{\alpha }^{-}}$, such that

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{1}{p^{+}}min\left\{{\parallel \eta \parallel }^{2p^{+}},{\parallel \eta \parallel }^{2p^{-}}\right\}+\frac{\mu }{q^{+}}min\left\{{\parallel \eta \parallel }^{2q^{+}},{\parallel \eta \parallel }^{2q^{-}}\right\}\hfill \\ \hfill & -\frac{\lambda }{e{h}^{-}}max\left\{C_{h^{+}+r^{+}}{\parallel \eta \parallel }^{h^{+}+r^{+}},C_{h^{-}+r^{-}}{\parallel \eta \parallel }^{h^{-}+r^{-}}\right\}\hfill \\ \hfill & -dmax\left\{C_{{\alpha }^{+}}{\parallel \eta \parallel }^{{\alpha }^{+}},C_{{\alpha }^{-}}{\parallel \eta \parallel }^{{\alpha }^{-}}\right\}-cmax\left\{{\parallel \eta \parallel }^{p^{+}},{\parallel \eta \parallel }^{p^{-}}\right\}\hfill \\ \hfill & -(\lambda C_{{\Omega }_1}+d)|\Omega |.\hfill \end{array}$$

If $\parallel \eta \parallel >1$, then

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{1}{p^{+}}{\parallel \eta \parallel }^{2p^{-}}+\frac{\mu }{q^{+}}min\left\{{\parallel \eta \parallel }^{2q^{+}},{\parallel \eta \parallel }^{2q^{-}}\right\}-\frac{\lambda C_{h^{+}+r^{+}}}{e{h}^{-}}{\parallel \eta \parallel }^{h^{+}+r^{+}}\hfill \\ \hfill & -d{C}_{\alpha }{\parallel \eta \parallel }^{{\alpha }^{+}}-{c\parallel \eta \parallel }^{p^{+}}-(\lambda C_{{\Omega }_1}+d)|\Omega |.\hfill \end{array}$$

Combined with the above analysis, we deduce that

Case 1: Utilizing that $2p^{-}>p^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda \le 0,$ there exists a positive constant R, provided at a sufficiently large size, such that

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{1}{p^{+}}{\parallel \eta \parallel }^{2p^{-}}-{c\parallel \eta \parallel }^{p^{+}}-d{C}_{\alpha }{\parallel \eta \parallel }^{{\alpha }^{+}}-d|\Omega |\ge 0,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\eta \in X_n,\mathrm{with}\parallel \eta \parallel =R.$

Case 2: Utilizing that $2p^{-}>p^{+}$, $p^{-}>q^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu <0$ and $\lambda \le 0,$ there exists a positive constant R, provided at a sufficiently large size, such that

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{1}{p^{+}}{\parallel \eta \parallel }^{2p^{-}}+\frac{\mu }{q^{+}}{\parallel \eta \parallel }^{2q^{+}}\hfill \\ \hfill & -{c\parallel \eta \parallel }^{p^{+}}-d{C}_{\alpha }{\parallel \eta \parallel }^{{\alpha }^{+}}-d|\Omega |\ge 0,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\eta \in X_n,\mathrm{with}\parallel \eta \parallel =R.$

Case 3: Utilizing that $2q^{-}>p^{+}$ and $q^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda \le 0,$ there exists a positive constant R, provided at a sufficiently large size, such that

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{\mu }{q^{+}}{\parallel \eta \parallel }^{2q^{-}}-{c\parallel \eta \parallel }^{p^{+}}-d{C}_{\alpha }{\parallel \eta \parallel }^{{\alpha }^{+}}-d|\Omega |\ge 0,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\eta \in X_n,\mathrm{with}\parallel \eta \parallel =R.$

Case 4: Utilizing that $2p^{-}>r^{+}+h^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda >0,$ there exists a positive constant R, provided at a sufficiently large size, such that

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{1}{p^{+}}{\parallel \eta \parallel }^{2p^{-}}-{c\parallel \eta \parallel }^{p^{+}}-d{C}_{\alpha }{\parallel \eta \parallel }^{{\alpha }^{+}}\hfill \\ \hfill & -\frac{\lambda C_{h^{+}+r^{+}}}{e{h}^{-}}{\parallel \eta \parallel }^{h^{+}+r^{+}}-(\lambda C_{{\Omega }_1}+d)|\Omega |\ge 0,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\eta \in X_n,\mathrm{with}\parallel \eta \parallel =R.$

Case 5: Utilizing that $2p^{-}>r^{+}+h^{+}$, $p^{-}>q^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu <0$ and $\lambda >0,$ there exists a positive constant R, provided at a sufficiently large size, such that

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{1}{p^{+}}{\parallel \eta \parallel }^{2p^{-}}+\frac{\mu }{q^{+}}{\parallel \eta \parallel }^{2q^{+}}-{c\parallel \eta \parallel }^{p^{+}}-d{C}_{\alpha }{\parallel \eta \parallel }^{{\alpha }^{+}}\hfill \\ \hfill & -\frac{\lambda C_{h^{+}+r^{+}}}{e{h}^{-}}{\parallel \eta \parallel }^{h^{+}+r^{+}}-(\lambda C_{{\Omega }_1}+d)|\Omega |\ge 0,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\eta \in X_n,\mathrm{with}\parallel \eta \parallel =R.$

Case 6: Utilizing that $2q^{-}>r^{+}+h^{+}$ and $q^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda >0,$ there exists a positive constant R, provided at a sufficiently large size, such that

$$\begin{array}{cc}\hfill \langle G,\eta \rangle \ge & \frac{\mu }{q^{+}}{\parallel \eta \parallel }^{2q^{-}}-{c\parallel \eta \parallel }^{p^{+}}-d{C}_{\alpha }{\parallel \eta \parallel }^{{\alpha }^{+}}\hfill \\ \hfill & -\frac{\lambda C_{h^{+}+r^{+}}}{e{h}^{-}}{\parallel \eta \parallel }^{h^{+}+r^{+}}-(\lambda C_{{\Omega }_1}+d)|\Omega |\ge 0,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\eta \in X_n,\mathrm{with}\parallel \eta \parallel =R.$

In the above six cases, G is continuous, so, in view of Lemma 3, problem (1) admits a approximate solution ${\eta }_n$ in $X_n\subset W_0$ with $\parallel {\eta }_n\parallel \le R.$ □

Corollary 1.

Assume that the conditions of Theorem 1 are satisfied, then the sequence ${\left\{{\eta }_n\right\}}_{n\ge 1}$ with ${\eta }_n\in X_n$ constructed in Theorem 1 is bounded in $W_0$.

Proof. 

If $\parallel {\eta }_n\parallel \le 1$ for all $n\in \mathbb{N}$, then the sequence ${\left\{{\eta }_n\right\}}_{n\in N}$ is bounded in $W_0$.

If $\parallel {\eta }_n\parallel >1$ for all $n\in \mathbb{N}$, with ${\eta }_n$ in place of $\phi$ in (14), we have

$$\begin{array}{cc}\hfill M_{p\left(x\right)}& \left({\delta }_{p\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,{\eta }_n\rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,{\eta }_n\rangle }_{q\left(x\right)}\hfill \\ \hfill & =\lambda {\int }_{\Omega }\left(|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right){\eta }_{n}dx+{\int }_{\Omega }g(x,{\eta }_n,\nabla {\eta }_n){\eta }_{n}dx.\hfill \end{array}$$

On the basis of condition $H_{g_1}$ and Lemma 5, it gives

$$\begin{array}{cc}\hfill \frac{1}{p^{+}}& {\left({\int }_{\Omega }{|\nabla {\eta }_n|}^{p\left(x\right)}dx\right)}^2+\frac{\mu }{q^{+}}{\left({\int }_{\Omega }{|\nabla {\eta }_n|}^{q\left(x\right)}dx\right)}^2\hfill \\ \hfill \le & c{\int }_{\Omega }|\nabla {\eta }_n{|}^{p\left(x\right)}dx+d{\int }_{\Omega }\left(\right|{\eta }_n{|}^{\alpha \left(x\right)}+1)dx+\lambda C_{{\Omega }_1}|\Omega |\hfill \\ \hfill & +\frac{\lambda }{e{h}^{-}}max\left\{C_{h^{+}+r^{+}}\parallel {\eta }_n{\parallel }^{h^{+}+r^{+}},C_{h^{-}+r^{-}}{\parallel {\eta }_n\parallel }^{h^{-}+r^{-}}\right\}.\hfill \end{array}$$

Case 1: Recalling that $2p^{-}>p^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda \le 0,$ and, by Lemmas 1 and 2, we deduce

$$\begin{array}{c}\hfill \frac{1}{p^{+}}\parallel {\eta }_n{\parallel }^{2p^{-}}\le c\parallel {\eta }_n{\parallel }^{p^{+}}+d{C}_{{\alpha }^{+}}\parallel {\eta }_n{\parallel }^{{\alpha }^{+}}+d|\Omega |.\end{array}$$

Case 2: Recalling that $2p^{-}>p^{+}$, $p^{-}>q^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu <0$ and $\lambda \le 0,$ and by Lemmas 1 and 2, we deduce

$$\begin{array}{c}\hfill \frac{1}{p^{+}}\parallel {\eta }_n{\parallel }^{2p^{-}}\le -\frac{\mu }{q^{+}}\parallel {\eta }_n{\parallel }^{2q^{+}}+c\parallel {\eta }_n{\parallel }^{p^{+}}+d{C}_{{\alpha }^{+}}\parallel {\eta }_n{\parallel }^{{\alpha }^{+}}+d|\Omega |.\end{array}$$

Case 3: Recalling that $2q^{-}>p^{+}$ and $q^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda \le 0,$ and by Lemmas 1 and 2, we deduce

$$\begin{array}{c}\hfill \frac{\mu }{q^{+}}\parallel {\eta }_n{\parallel }^{2q^{-}}\le c\parallel {\eta }_n{\parallel }^{p^{+}}+d{C}_{\alpha }\parallel {\eta }_n{\parallel }^{{\alpha }^{+}}+d|\Omega |.\end{array}$$

Case 4: Recalling that $2p^{-}>r^{+}+h^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda >0,$ and by Lemmas 1 and 2, we deduce

$$\begin{array}{cc}\hfill \frac{1}{p^{+}}{\parallel {\eta }_n\parallel }^{2p^{-}}\le & c\parallel {\eta }_n{\parallel }^{p^{+}}+d{C}_{\alpha }{\parallel {\eta }_n\parallel }^{{\alpha }^{+}}\hfill \\ \hfill & +\frac{\lambda C_{h^{+}+r^{+}}}{e{h}^{-}}\parallel {\eta }_n{\parallel }^{h^{+}+r^{+}}+(\lambda C_{{\Omega }_1}+d)|\Omega |.\hfill \end{array}$$

Case 5: Recalling that $2p^{-}>r^{+}+h^{+}$, $p^{-}>q^{+}$ and $p^{-}>{\alpha }^{+}$ with $\mu <0$ and $\lambda >0,$ and by Lemmas 1 and 2, we deduce

$$\begin{array}{cc}\hfill \frac{1}{p^{+}}{\parallel {\eta }_n\parallel }^{2p^{-}}\le & -\frac{\mu }{q^{+}}\parallel {\eta }_n{\parallel }^{2q^{+}}+c\parallel {\eta }_n{\parallel }^{p^{+}}+d{C}_{\alpha }{\parallel {\eta }_n\parallel }^{{\alpha }^{+}}\hfill \\ \hfill & +\frac{\lambda C_{h^{+}+r^{+}}}{e{h}^{-}}\parallel {\eta }_n{\parallel }^{h^{+}+r^{+}}+(\lambda C_{{\Omega }_1}+d)|\Omega |.\hfill \end{array}$$

Case 6: Recalling that $2p^{-}>r^{+}+h^{+}$ and $q^{-}>{\alpha }^{+}$ with $\mu \ge 0$ and $\lambda >0,$ and by Lemmas 1 and 2, we deduce

$$\begin{array}{cc}\hfill \frac{\mu }{q^{+}}{\parallel {\eta }_n\parallel }^{2q^{-}}\le & c\parallel {\eta }_n{\parallel }^{p^{+}}-d{C}_{\alpha }{\parallel {\eta }_n\parallel }^{{\alpha }^{+}}\hfill \\ \hfill & +\frac{\lambda C_{h^{+}+r^{+}}}{e{h}^{-}}\parallel {\eta }_n{\parallel }^{h^{+}+r^{+}}+(\lambda C_{{\Omega }_1}+d)|\Omega |.\hfill \end{array}$$

In the above six cases, we conclude that the sequence ${\left\{{\eta }_n\right\}}_{n\ge 1}$ is bounded in $W_0.$ □

4. Existence of Weak Solutions

In this section, our interest is devoted to the existence of weak solutions for problem (1). The following are the main results of this section.

Theorem 2.

Assume that conditions $H_m$, $H_{pq}$, and $H_{g_2}$ are satisfied, then, for all $\mu >0,$

• if $2p^{-}>{\varphi }^{+}+1$ and $2p^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+}$, problem (1) admits at least one weak solution with $\lambda \le 0.$
• if $2p^{-}>h^{+}+r^{+}$, $2p^{-}>{\varphi }^{+}+1$ and $2p^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+},$ problem (1) admits at least one weak solution with $\lambda >0.$
• if $2q^{-}>{\varphi }^{+}+1$ and $2q^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+},$ problem (1) admits at least one weak solution with $\lambda \le 0.$
• if $2q^{-}>h^{+}+r^{+}$, $2q^{-}>{\varphi }^{+}+1$ and $2q^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+},$ problem (1) admits at least one weak solution with $\lambda >0.$

Corollary 2.

Assume that the conditions of Theorem 2 are satisfied; then, the sequence ${\left\{{\eta }_n\right\}}_{n\ge 1}$ with ${\eta }_n\in X_n$ is bounded in $W_0$.

Proof. 

The proof is similar to Corollary 1, which we omit. □

To prove Theorems 2, we use the Brezis theorem for pseudomonotone operators in the separable reflexive space (see (Theorem 27.A [38]). Let us define the operator $T:W_0\to W_0^{*}$ as

$$\begin{array}{cc}\hfill \langle T\eta ,\phi \rangle =& M_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\langle \eta ,\phi \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\langle \eta ,\phi \rangle }_{q\left(x\right)}\hfill \\ \hfill & -\lambda {\int }_{\Omega }\left({\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right)\phi dx-{\int }_{\Omega }g(x,\eta ,\nabla \eta )\phi dx,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\eta ,\phi \in W_0.$

Lemma 6.

Assume that the conditions of Theorem 2 are satisfied; then, the operator T is bounded.

Proof. 

Let $\eta \in W_0$ be fixed and denote by ${\Phi }_{\eta }$ the linear functional on $W_0$, defined as

$$\begin{array}{c}\hfill {\Phi }_{\eta }\left(\phi \right)={\int }_{\Omega }{|\nabla \eta |}^{\upsilon \left(x\right)-2}\nabla \eta \nabla \phi dx,\end{array}$$

for any $\phi \in W_0$ and $\upsilon \left(x\right)\in \left\{p\right(x),q(x\left)\right\}$. By Hölder inequality,

$$\begin{array}{c}\hfill |{\Phi }_{\eta }\left(\phi \right)|\le \parallel \eta \parallel \parallel \phi \parallel ,\mathrm{for}\mathrm{all}\eta ,\phi \in W_0.\end{array}$$

(16)

Obviously, ${\Phi }_{\eta }\left(\phi \right)$ is bounded. From the hypothesis $H_m$ and Proposition 1, there exist some constants $C_{{\upsilon }_1},C_{{\upsilon }_2}>0$ such that

$$\begin{array}{c}\hfill 0<C_{{\upsilon }_1}\le M_{\upsilon \left(x\right)}\left({\delta }_{\upsilon \left(x\right)}\left(\eta \right)\right)\le C_{{\upsilon }_2},\end{array}$$

which, together with (16), there exists a constant $C_{\upsilon \left(x\right)}>0$ such that

$$\begin{array}{c}\hfill |M_{\upsilon \left(x\right)}\left({\delta }_{\upsilon \left(x\right)}\left(\eta \right)\right){\langle \eta ,\phi \rangle }_{\upsilon \left(x\right)}|\le C_{\upsilon \left(x\right)}.\end{array}$$

(17)

In fact, by a simple calculation for the logarithmic nonlinear term, we deduce

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left|{\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx=& {\int }_{{\Omega }_1}{\left|{\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx+{\int }_{{\Omega }_2}{\left|{\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx\hfill \\ \hfill \le & C_{{\Omega }_1}|\Omega |+{\int }_{{\Omega }_2}{\left|{\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx.\hfill \end{array}$$

Since $r^{+}<p^{*}\left(x\right)$, then, by using the continuous embedding $L^{p^{*}\left(x\right)}(\Omega )\hookrightarrow L^{r^{+}}(\Omega )$ and combining Lemma 4, we deduce

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left|{\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx& \le C_{{\Omega }_1}|\Omega |+{\int }_{\Omega }{\left|\eta \right|}^{r^{+}}dx\hfill \\ \hfill & \le C_{{\Omega }_1}|\Omega |+C_{{\Omega }_2}{\parallel \eta \parallel }_{p^{*}\left(x\right)}.\hfill \end{array}$$

(18)

where $C_{{\Omega }_2}>0$. Notice that the relation (18) implies that

$$\begin{array}{c}\hfill {∥{\left|\eta \right|}^{r\left(x\right)-1}ln\left|\eta \right|∥}_{L^{\frac{r^{+}}{r^{+}-1}}(\Omega )}\le C_{\frac{r^{+}}{r^{+}-1}},\end{array}$$

where $C_{\frac{r^{+}}{r^{+}-1}}>0$. Using the Hölder inequality and taking into account the embeddings, for any $\phi \in W_0$ with $\parallel \phi \parallel \le 1$,

$$\begin{array}{cc}\hfill |{\int }_{\Omega }{\phi \left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|dx|& \le {\parallel \phi \parallel }_{L^{r^{+}}(\Omega )}{∥{\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|∥}_{L^{\frac{r^{+}}{r^{+}-1}}(\Omega )}\le C_{\frac{r^{+}}{r^{+}-1}}.\hfill \end{array}$$

(19)

From hypothesis ${\mathcal{G}}_1$ and Jensen’s inequality, for all $\eta \in X$, we have

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\left|g(x,\eta ,\nabla \eta )\right|}^{p^{\prime }\left(x\right)}dx\hfill \\ \hfill \le & {\int }_{\Omega }{\left[\left|h\left(x\right)\right|+{\left|a\right|\eta |}^{\varphi \left(x\right)}{|+|b|\nabla \eta |}^{\frac{\psi \left(x\right)}{p^{\prime }\left(x\right)}}|\right]}^{p^{\prime }\left(x\right)}dx\hfill \\ \hfill \le & 3^{{\left(q^{\prime }\right)}^{+}-1}\left[{\int }_{\Omega }{\left|h\left(x\right)\right|}^{p^{\prime }\left(x\right)}dx+{\int }_{\Omega }{\left|a\right|\eta |}^{\varphi \left(x\right)}{|}^{p^{\prime }\left(x\right)}dx+{\int }_{\Omega }{\left|b\right|\nabla \eta |}^{\frac{\psi \left(x\right)}{p^{\prime }\left(x\right)}}{|}^{p^{\prime }\left(x\right)}dx\right]\hfill \\ \hfill \le & C_{p^{\prime }}\left[{\int }_{\Omega }{\left|h\left(x\right)\right|}^{p^{\prime }\left(x\right)}dx+{\int }_{\Omega }{\left|\eta \right|}^{p^{\prime }\left(x\right)\varphi \left(x\right)}dx+{\int }_{\Omega }{|\nabla \eta |}^{\psi \left(x\right)}dx\right],\hfill \end{array}$$

(20)

where $C_{p^{\prime }}=3^{{\left(q^{\prime }\right)}^{+}-1}max\left\{1,a^{{\left(p^{\prime }\right)}^{-}},a^{{\left(p^{\prime }\right)}^{+}},b^{{\left(p^{\prime }\right)}^{-}},b^{{\left(p^{\prime }\right)}^{+}}\right\}.$ It follows from (20) and Proposition 1 that we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left|g(x,\eta ,\nabla \eta )\right|}^{p^{\prime }\left(x\right)}dx\le & C_{p^{\prime }}\left[3+{\left|h\right|}_{p^{\prime }}^{{\left(p^{\prime }\right)}^{+}}+{\left|\eta \right|}_{p^{\prime }\varphi }^{{\left(p^{\prime }\varphi \right)}^{+}}+{\parallel \eta \parallel }^{{\psi }^{+}}\right]\hfill \\ \hfill \le & C_{p^{\prime }}\left[3+{\left|h\right|}_{p^{\prime }}^{{\left(p^{\prime }\right)}^{+}}+C_{p^{\prime }\varphi }^{{\left(p^{\prime }\varphi \right)}^{+}}{\parallel \eta \parallel }_{p^{\prime }\varphi }^{{\left(p^{\prime }\varphi \right)}^{+}}+{\parallel \eta \parallel }^{{\psi }^{+}}\right].\hfill \end{array}$$

Hence, invoking Proposition 1, we infer

$$\begin{array}{cc}\hfill {\left|g(x,\eta ,\nabla \eta )\right|}_{p^{\prime }}\le & {\left\{1+C_{p^{\prime }}{[3+|h|}_{p^{\prime }}^{{\left(p^{\prime }\right)}^{+}}+C_{p^{\prime }\varphi }^{{\left(p^{\prime }\varphi \right)}^{+}}{\parallel \eta \parallel }_{p^{\prime }\varphi }^{{\left(p^{\prime }\varphi \right)}^{+}}+{\parallel \eta \parallel }^{{\psi }^{+}}]\right\}}^{\frac{1}{{\left(p^{\prime }\right)}^{-}}}.\hfill \end{array}$$

(21)

Utilizing Lemma 1and taking into account the embeddings, for all $\phi \in W_0$ with $\parallel \phi \parallel \le 1,$

$$\begin{array}{c}\hfill \left|{\int }_{\Omega }g(x,\eta ,\nabla \eta )\phi dx\right|\le {2\left|g(x,\eta ,\nabla \eta )\right|}_{p^{\prime }}{\left|\phi \right|}_p\le 2{\left|g(x,\eta ,\nabla \eta )\right|}_{p^{\prime }}.\end{array}$$

(22)

Thus, it follows from these estimates (17), (19), and (22) that we easily determine the boundedness of T. □

Lemma 7.

Assume that the conditions of Theorem 2 are satisfied; then, the operator T is demicontinuous.

Proof. 

Assuming that ${\eta }_n\to \eta$ in $W_0$, we show that $T{\eta }_n\to T\eta$ in $W_0^{*}$, that is,

$$\begin{array}{cc}\hfill M_{p\left(x\right)}& \left({\delta }_{p\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,\phi \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,\phi \rangle }_{q\left(x\right)}\hfill \\ \hfill & -\lambda {\int }_{\Omega }\left(|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right)\phi dx-{\int }_{\Omega }g(x,{\eta }_n,\nabla {\eta }_n)\phi dx\hfill \\ \hfill & \to M_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\langle \eta ,\phi \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\langle \eta ,\phi \rangle }_{q\left(x\right)}\hfill \\ \hfill & -\lambda {\int }_{\Omega }\left({\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right)\phi dx-{\int }_{\Omega }g(x,\eta ,\nabla \eta )\phi dx.\hfill \end{array}$$

(23)

Since ${\eta }_n\to \eta$ in $W_0$, up to a subsequence, we have

$$\begin{array}{c}\hfill {\eta }_n\to \eta \mathrm{and}\nabla {\eta }_n\to \nabla \eta ,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .\end{array}$$

(24)

Thus, we have

$$\left|\right|\nabla {\eta }_n{|}^{\upsilon \left(x\right)-2}\nabla {\eta }_n{|}^{p^{\prime }\left(x\right)}\le \parallel {\eta }_n\parallel (1+\parallel {\eta }_n{\parallel }_{p^{\prime }}^{{\upsilon }^{+}-2}),\upsilon \in \{p\left(x\right),q\left(x\right)\}.$$

(25)

which imply that $\left\{\right|\nabla {\eta }_n{|}^{\upsilon \left(x\right)-2}\nabla {\eta }_n\}$ are bounded in $L^{p^{\prime }}(\Omega )$.

For $\upsilon \left(x\right)\in \left\{p\right(x),q(x\left)\right\},$ we obtain

$$\left\{\begin{array}{c}|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln|{\eta }_n{|\to |\eta |}^{r\left(x\right)-2}\eta ln\left|\eta \right|,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\hfill \\ |\nabla {\eta }_n{|}^{\upsilon \left(x\right)-2}\nabla {\eta }_n\to {|\nabla \eta |}^{\upsilon \left(x\right)-2}\nabla \eta ,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,\hfill \\ g(x,{\eta }_n,\nabla {\eta }_n)\to g(x,\eta ,\nabla \eta ),\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .\hfill \end{array}\right.$$

(26)

Moreover, the boundedness of $\left\{{\eta }_n\right\}$ in $W_0$ and (22) imply that $\left\{g(x,{\eta }_n,\nabla {\eta }_n)\right\}$ are bounded in $L^{p^{\prime }}(\Omega )$, and (19) implies that $\left\{\right|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\}$ are bounded in $L^{r^{\prime }}(\Omega )$. Thanks to (17), (24), and (26), combined with $H_m$ and Proposition 1, we obtain

$$\begin{array}{c}\hfill M_{\upsilon \left(x\right)}\left({\delta }_{\upsilon \left(x\right)}\left({\eta }_n\right)\right){\langle {\eta }_n,\phi \rangle }_{\upsilon \left(x\right)}\to M_{\upsilon \left(x\right)}\left({\delta }_{\upsilon \left(x\right)}\left(\eta \right)\right){\langle \eta ,\phi \rangle }_{\upsilon \left(x\right)}.\end{array}$$

(27)

Now, we show that the following conclusion holds:

$$\begin{array}{c}\hfill {\int }_{\Omega }g(x,{\eta }_n,\nabla {\eta }_n)\phi dx\to {\int }_{\Omega }g(x,\eta ,\nabla \eta )\phi dx.\end{array}$$

(28)

Let $g(x,{\eta }_n,\nabla {\eta }_n),g(x,\eta ,\nabla \eta )\in L^{p^{\prime }}(\Omega )$, and

$$\begin{array}{c}\hfill E\left(N\right)=\left\{x\in :|g(x,{\eta }_n,\nabla {\eta }_n)-g(x,\eta ,\nabla \eta )|\le 1,\mathrm{for}\mathrm{all}n\ge N\right\}.\end{array}$$

Since meas$\left(E\right(N\left)\right)\to$ meas$(\Omega )$ as $N\to \infty$, and setting

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathcal{N}}=\left\{{\Psi }_N\in L^{p^{″}\left(x\right)}(\Omega ):{\Psi }_N\equiv 0\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \backslash E\left(N\right)\right\}.\end{array}$$

First, we prove that ${\mathcal{F}}_{\mathcal{N}}$ is dense in $L^{p^{″}\left(x\right)}(\Omega )$. Let $f\in L^{p^{″}\left(x\right)}(\Omega )$ and

$$\begin{array}{c}\hfill f_N\left(x\right)=\left\{\begin{array}{c}f\left(x\right)\mathrm{if} x\in \left(E\right(N\left)\right),\hfill \\ 0\mathrm{if} x\in \Omega \backslash \left(E\right(N\left)\right).\hfill \end{array}\right.\end{array}$$

Then,

$$\begin{array}{cc}\hfill {\varrho }_{p^{″}\left(x\right)}\left(f_N\left(x\right)-f\left(x\right)\right)=& {\int }_{E\left(N\right)}|f_N{\left(x\right)-f\left(x\right)|}^{p^{″}\left(x\right)}dx+{\int }_{\Omega \backslash E\left(N\right)}{|f_N\left(x\right)-f\left(x\right)|}^{p^{″}\left(x\right)}dx\hfill \\ \hfill =& {\int }_{\Omega \backslash E\left(N\right)}{\left|f\left(x\right)\right|}^{p^{″}\left(x\right)}dx\hfill \\ \hfill =& {\int }_{\Omega }{\left|f\left(x\right)\right|}^{p^{″}\left(x\right)}{\chi }_{\Omega \backslash E\left(N\right)}dx.\hfill \end{array}$$

Taking ${\Phi }_N={\left|f\left(x\right)\right|}^{p^{″}\left(x\right)}{\chi }_{\Omega \backslash E\left(N\right)}$ for almost every x in $\Omega$, we have

$$\begin{array}{c}\hfill {\Phi }_N\to 0\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \mathrm{and}|{\Phi }_N{|\le |f|}^{p^{″}\left(x\right)}.\end{array}$$

Utilizing the dominated convergence theorem, we infer

$$\begin{array}{c}\hfill {\varrho }_{p^{″}\left(x\right)}(f_N\left(x\right)-f\left(x\right))\to 0\mathrm{as}N\to \infty ,\end{array}$$

hence $f_N\to f$ in $L^{p^{″}\left(x\right)}(\Omega )$. Thus, ${\mathcal{F}}_{\mathcal{N}}$ is dense in $L^{p^{″}\left(x\right)}(\Omega )$.

Next, $\mathrm{for}\mathrm{all}\phi \in {\mathcal{F}}_{\mathcal{N}}$, let us show that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{\Omega }\left(g(x,{\eta }_n,\nabla {\eta }_n)-g(x,\eta ,\nabla \eta )\right)\phi \left(x\right)dx=0.\end{array}$$

(29)

Since $\phi \equiv 0$ in $\Omega \backslash E\left(N\right)$, it suffices to prove that

$$\begin{array}{c}\hfill {\int }_{E\left(N\right)}\left(g(x,{\eta }_n,\nabla {\eta }_n)-g(x,\eta ,\nabla \eta )\right)\phi \left(x\right)dx\to 0\mathrm{as}n\to \infty .\end{array}$$

Let ${\phi }_n=\phi (g(x,{\eta }_n,\nabla {\eta }_n)-g(x,\eta ,\nabla \eta )).$ Since $\left|\right(g(x,{\eta }_n,\nabla {\eta }_n-g(x,\eta ,\nabla \eta ))\phi \left(x\right)|\le \phi \left(x\right)$ a.e. in $E\left(N\right)$ and ${\phi }_n\to 0$ a.e. in $\Omega$, thanks to the dominated convergence theorem, we deduce ${\phi }_n\to 0$ in $L^1(\Omega )$, which implies that (29) holds.

It follows from the density of ${\mathcal{F}}_{\mathcal{N}}$ in $L^{p^{″}\left(x\right)}(\Omega )$ that we deduce

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{\Omega }g(x,{\eta }_n,\nabla {\eta }_n)\phi \left(x\right)dx=\underset{n\to \infty }{lim}{\int }_{\Omega }g(x,\eta ,\nabla \eta )\phi \left(x\right)dx,\end{array}$$

$\mathrm{for}\mathrm{all}\phi \in {\mathcal{F}}_{\mathcal{N}}$, which implies that (28) holds.

Using the same discussion as above, one can conclude that

$$\begin{array}{c}\hfill {\int }_{\Omega }\left(|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right)\phi dx\to {\int }_{\Omega }\left({\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right)\phi dx.\end{array}$$

(30)

As a result, it follows from (27), (28), and (30) that (23) holds, that is, the operator T is demicontinuous. □

Lemma 8.

Assume that the conditions of Theorem 2 are satisfied; then, for all $\mu >0$, the operator T is coercive.

Proof. 

First, $\mathrm{for}\mathrm{all}\eta \in W_0,$ we note that

$$\begin{array}{cc}\hfill \langle T\eta ,\eta \rangle =& M_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\langle \eta ,\eta \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\langle \eta ,\eta \rangle }_{q\left(x\right)}\hfill \\ \hfill & -\lambda {\int }_{\Omega }\left({\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right)\eta dx-{\int }_{\Omega }g(x,\eta ,\nabla \eta )\eta dx.\hfill \end{array}$$

(31)

To estimate the first and second integral terms, we deduce

$$\begin{array}{cc}\hfill M_{p\left(x\right)}& \left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\langle \eta ,\eta \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\langle \eta ,\eta \rangle }_{q\left(x\right)}\hfill \\ \hfill & \ge \frac{1}{p^{+}}{\left({\int }_{\Omega }{|\nabla \eta |}^{p\left(x\right)}dx\right)}^2+\frac{\mu }{q^{+}}{\left({\int }_{\Omega }{|\nabla \eta |}^{q\left(x\right)}dx\right)}^2\hfill \\ \hfill & \ge \frac{1}{p^{+}}min\left\{{\parallel \eta \parallel }^{2p^{+}},{\parallel \eta \parallel }^{2p^{-}}\right\}+\frac{\mu }{q^{+}}min\left\{{\parallel \eta \parallel }^{2q^{+}},{\parallel \eta \parallel }^{2q^{-}}\right\}\hfill \\ \hfill & \ge \frac{1}{p^{+}}\left\{{\parallel \eta \parallel }^{2p^{-}}-1\right\}+\frac{\mu }{q^{+}}\left\{{\parallel \eta \parallel }^{2q^{-}}-1\right\}.\hfill \end{array}$$

(32)

To estimate the third integral term, let ${\Omega }_1=\{x\in \Omega :|\eta \left(x\right)|\le 1\}$ and ${\Omega }_2=\{x\in \Omega :|\eta \left(x\right)|\ge 1\}$; then,

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx={\int }_{{\Omega }_1}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx+{\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx.\end{array}$$

Using Lemma 4 with $h\left(x\right)+r\left(x\right)\le h^{+}+r^{+}<p^{*}\left(x\right)$, we deduce

$$\begin{array}{c}\hfill {\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx\le \frac{1}{e{h}^{-}}{\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)+h\left(x\right)}dx\le \frac{1}{e{h}^{-}}\left({\left|\eta \right|}_{h\left(x\right)+r\left(x\right)}^{h^{+}+r^{+}}+1\right),\end{array}$$

in view of Lemma 2, there exist some constants $C_{h^{+}+r^{+}}>0$ and $C_{h^{-}+r^{-}}>0$ such that

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_2}{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx\le & \frac{1}{e{h}^{-}}{C}_{h^{+}+r^{+}}\left({\parallel \eta \parallel }^{h^{+}+r^{+}}+1\right).\hfill \end{array}$$

This implies that

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left|\eta \right|}^{r\left(x\right)}ln\left|\eta \right|dx\le C_{{\Omega }_1}|\Omega |+\frac{1}{e{h}^{-}}{C}_{h^{+}+r^{+}}\left({\parallel \eta \parallel }^{h^{+}+r^{+}}+1\right),\end{array}$$

(33)

where $|\Omega |$ denotes the Lebesgue measure of $\Omega$ and $C_{{\Omega }_1}>0$. This yields the stated conclusion.

To estimate the fourth integral term, we deduce from $H_{g_2}$, the Hölder-type inequality, and Proposition 1 that

$$\begin{array}{cc}\hfill {\int }_{\Omega }g(x,\eta ,\nabla \eta )\eta dx\le & {\int }_{\Omega }\left(h\left(x\right)+{a\left|\eta \right|}^{\varphi \left(x\right)}+b{|\nabla \eta |}^{\frac{\psi \left(x\right)}{p^{\prime }\left(x\right)}}\right)\eta dx\hfill \\ \hfill \le & {\left|h\right|}_{p^{\prime }}{\left|\eta \right|}_p+a\left({\left|\eta \right|}_{\varphi \left(x\right)+1}^{{\varphi }^{+}+1}+1\right)+b\left({\parallel \eta \parallel }^{{\left(\frac{\psi }{p^{\prime }}\right)}^{+}}+1\right){\left|\eta \right|}_p\hfill \\ \hfill \le & C_p{\left|h\right|}_{p^{\prime }}\parallel \eta \parallel +a{C}_{\varphi \left(x\right)+1}^{{\varphi }^{+}+1}\left({\parallel \eta \parallel }^{{\varphi }^{+}+1}+1\right)+b{C}_p\parallel \eta \parallel \left({\parallel \eta \parallel }^{{\left(\frac{\psi }{p^{\prime }}\right)}^{+}}+1\right).\hfill \end{array}$$

(34)

It follows (32), (33), and (34) that

$$\begin{array}{cc}\hfill \langle T\eta ,\eta \rangle \ge & \frac{1}{p^{+}}\left\{{\parallel \eta \parallel }^{2p^{-}}-1\right\}+\frac{\mu }{q^{+}}\left\{{\parallel \eta \parallel }^{2q^{-}}-1\right\}\hfill \\ \hfill & -\lambda C_{{\Omega }_1}|\Omega |-\frac{\lambda }{e{h}^{-}}{C}_{h^{+}+r^{+}}\left({\parallel \eta \parallel }^{h^{+}+r^{+}}+1\right)\hfill \\ \hfill & -C_p{\left|h\right|}_{p^{\prime }}\parallel \eta \parallel -a{C}_{\varphi \left(x\right)+1}^{{\varphi }^{+}+1}\left({\parallel \eta \parallel }^{{\varphi }^{+}+1}+1\right)-b{C}_p\parallel \eta \parallel \left({\parallel \eta \parallel }^{{\left(\frac{\psi }{p^{\prime }}\right)}^{+}}+1\right).\hfill \end{array}$$

(35)

Case 1: Utilizing that $2p^{-}>{\varphi }^{+}+1$ and $2p^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+}$ with $\lambda \le 0,$$\mathrm{for}\mathrm{all}\eta \in W_0,$ such that

$$\begin{array}{cc}\hfill \langle T\eta ,\eta \rangle \ge & \frac{1}{p^{+}}\left\{{\parallel \eta \parallel }^{2p^{-}}-1\right\}-C_p{\left|h\right|}_{p^{\prime }}\parallel \eta \parallel \hfill \\ \hfill & -a{C}_{\varphi \left(x\right)+1}^{{\varphi }^{+}+1}\left({\parallel \eta \parallel }^{{\varphi }^{+}+1}+1\right)-b{C}_p\parallel \eta \parallel \left({\parallel \eta \parallel }^{{\left(\frac{\psi }{p^{\prime }}\right)}^{+}}+1\right).\hfill \end{array}$$

Case 2: Utilizing that $2p^{-}>h^{+}+r^{+}$, $2p^{-}>{\varphi }^{+}+1$ and $2p^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+}$ with $\lambda >0,$$\mathrm{for}\mathrm{all}\eta \in W_0,$ such that

$$\begin{array}{cc}\hfill \langle T\eta ,\eta \rangle \ge & \frac{1}{p^{+}}\left\{{\parallel \eta \parallel }^{2p^{-}}-1\right\}-\lambda C_{{\Omega }_1}|\Omega |-\frac{\lambda }{e{h}^{-}}{C}_{h^{+}+r^{+}}\left({\parallel \eta \parallel }^{h^{+}+r^{+}}+1\right)\hfill \\ \hfill & -C_p{\left|h\right|}_{p^{\prime }}\parallel \eta \parallel -a{C}_{\varphi \left(x\right)+1}^{{\varphi }^{+}+1}\left({\parallel \eta \parallel }^{{\varphi }^{+}+1}+1\right)-b{C}_p\parallel \eta \parallel \left({\parallel \eta \parallel }^{{\left(\frac{\psi }{p^{\prime }}\right)}^{+}}+1\right).\hfill \end{array}$$

Case 3: Utilizing that $2q^{-}>{\varphi }^{+}+1$ and $2q^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+}$ with $\lambda \le 0,$$\mathrm{for}\mathrm{all}\eta \in W_0,$ such that

$$\begin{array}{cc}\hfill \langle T\eta ,\eta \rangle \ge & \frac{\mu }{q^{+}}\left\{{\parallel \eta \parallel }^{2q^{-}}-1\right\}-C_p{\left|h\right|}_{p^{\prime }}\parallel \eta \parallel \hfill \\ \hfill & -a{C}_{\varphi \left(x\right)+1}^{{\varphi }^{+}+1}\left({\parallel \eta \parallel }^{{\varphi }^{+}+1}+1\right)-b{C}_p\parallel \eta \parallel \left({\parallel \eta \parallel }^{{\left(\frac{\psi }{p^{\prime }}\right)}^{+}}+1\right).\hfill \end{array}$$

Case 4: Utilizing that $2q^{-}>h^{+}+r^{+}$, $2q^{-}>{\varphi }^{+}+1$ and $2q^{-}>{\left(\frac{\psi }{p^{\prime }}\right)}^{+}$ with $\lambda >0,$$\mathrm{for}\mathrm{all}\eta \in W_0,$ such that

$$\begin{array}{cc}\hfill \langle T\eta ,\eta \rangle \ge & \frac{\mu }{q^{+}}\left\{{\parallel \eta \parallel }^{2q^{-}}-1\right\}-\lambda C_{{\Omega }_1}|\Omega |-\frac{\lambda }{e{h}^{-}}{C}_{h^{+}+r^{+}}\left({\parallel \eta \parallel }^{h^{+}+r^{+}}+1\right)\hfill \\ \hfill & -C_p{\left|h\right|}_{p^{\prime }}\parallel \eta \parallel -a{C}_{\varphi \left(x\right)+1}^{{\varphi }^{+}+1}\left({\parallel \eta \parallel }^{{\varphi }^{+}+1}+1\right)-b{C}_p\parallel \eta \parallel \left({\parallel \eta \parallel }^{{\left(\frac{\psi }{p^{\prime }}\right)}^{+}}+1\right).\hfill \end{array}$$

In the above four cases, we deduce the coerciveness of T from (35) as $\parallel \eta \parallel \to \infty .$ □

Lemma 9.

Assume that the conditions of Theorem 2 are satisfied, then T is an $\left(S_{+}\right)$-type operator.

Proof. 

Let $\left\{{\eta }_n\right\}\in W_0$ be such that ${\eta }_n\rightharpoonup \eta$ in $W_0$ as $n\to \infty$ and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}\langle T{\eta }_n-T\eta ,{\eta }_n-\eta \rangle \le 0.\end{array}$$

First, note that

$$\begin{array}{cc}\hfill \langle T{\eta }_n,{\eta }_n-\eta \rangle =& M_{p\left(x\right)}\left({\delta }_{p\left(x\right)}\left(\eta \right)\right){\langle \eta ,{\eta }_n-\eta \rangle }_{p\left(x\right)}+\mu M_{q\left(x\right)}\left({\delta }_{q\left(x\right)}\left(\eta \right)\right){\langle \eta ,{\eta }_n-\eta \rangle }_{q\left(x\right)}\hfill \\ \hfill & -\lambda {\int }_{\Omega }\left({\left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|\right)({\eta }_n-\eta )dx-{\int }_{\Omega }g(x,\eta ,\nabla \eta )({\eta }_n-\eta )dx.\hfill \end{array}$$

(36)

Going if necessary up to a subsequence, we suppose there exists $\eta \in W_0$ such that

$$\begin{array}{cc}\hfill & {\eta }_n\rightharpoonup \eta ,\mathrm{weakly}\mathrm{in}{W}_0,\hfill \\ \hfill & {\eta }_n\to \eta ,\mathrm{strongly}\mathrm{in}{L}^{p\left(x\right)}(\Omega ),\hfill \\ \hfill & {\eta }_n\to \eta ,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .\hfill \end{array}$$

(37)

Indeed, by a simple calculation for the logarithmic nonlinear term, we deduce

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\left||{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx\hfill \\ \hfill & ={\int }_{{\Omega }_1}{\left||{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx+{\int }_{{\Omega }_2}{\left||{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx\hfill \\ \hfill & \le C_{{\Omega }_1}|\Omega |+{\int }_{{\Omega }_2}{\left||{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx.\hfill \end{array}$$

Since $r^{+}<p^{*}\left(x\right)$, then, by using the continuous embedding $L^{p^{*}\left(x\right)}(\Omega )\hookrightarrow L^{r^{+}}(\Omega )$ and combining Lemma 4, we deduce

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left||{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|\right|}^{\frac{r^{+}}{r^{+}-1}}dx& \le C_{{\Omega }_1}|\Omega |+{\int }_{\Omega }{\left|{\eta }_n\right|}^{r^{+}}dx\hfill \\ \hfill & \le C_{{\Omega }_1}|\Omega |+C_{{\Omega }_2}{\parallel {\eta }_n\parallel }_{p^{*}\left(x\right)}.\hfill \end{array}$$

(38)

where $C_{{\Omega }_2}>0$. In conjunction with Hölder’s inequality, we obtain

$$\begin{array}{cc}\hfill \left|{\int }_{\Omega }({\eta }_n-\eta )|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|dx\right|& \le \parallel {\eta }_n{-\eta \parallel }_{L^{r^{+}}(\Omega )}{∥|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|∥}_{L^{\frac{r^{+}}{r^{+}-1}}(\Omega )}.\hfill \end{array}$$

(39)

Therefore, it follows from (37), (38) and (39) that

$$\begin{array}{c}\hfill \left|{\int }_{\Omega }({\eta }_n-\eta )|{\eta }_n{|}^{r\left(x\right)-2}{\eta }_{n}ln\left|{\eta }_n\right|dx\right|\to 0,\mathrm{as}n\to \infty .\end{array}$$

(40)

In the same fashion, utilizing Lemma 1, we have

$$\begin{array}{c}\hfill |{\int }_{\Omega }g(x,\eta ,\nabla \eta )({\eta }_n-\eta )dx|\le 2|g(x,\eta ,\nabla \eta ){|}_{p^{\prime }}{\left|({\eta }_n-\eta )\right|}_p.\end{array}$$

By the boundedness of $\left\{{\eta }_n\right\}\in W_0$ and (37), we infer from the inequality above and the preceding estimate (21) that

$$\begin{array}{c}\hfill \left|{\int }_{\Omega }g(x,\eta ,\nabla \eta )({\eta }_n-\eta )dx\right|\to 0,\mathrm{as}n\to \infty .\end{array}$$

(41)

If ${\eta }_n\rightharpoonup \eta$ in $W_0$ and ${lim\; sup}_{n\to \infty }\langle T{\eta }_n-T\eta ,{\eta }_n-\eta \rangle \le 0,$ as a consequence

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\langle T{\eta }_n,{\eta }_n-\eta \rangle =\underset{n\to \infty }{lim}\langle T{\eta }_n-T\eta ,{\eta }_n-\eta \rangle =0.\end{array}$$

(42)

By (40), (41), (42), and $H_m$, for $\upsilon \in \left\{p\right(x),q(x\left)\right\}$, $\mathrm{as}n\to \infty$, we deduce

$$\begin{array}{cc}\hfill {\int }_{\Omega }\left(\right|\nabla {\eta }_n{|}^{\upsilon \left(x\right)-2}\nabla {\eta }_n-{|\nabla \eta |}^{\upsilon \left(x\right)-2}\nabla \eta )(\nabla {\eta }_n-\nabla \eta )dx& \to 0.\hfill \end{array}$$

Using the following Simon inequalities

$$|u_1-u_2{|}^{\tau }\le \left\{\begin{array}{c}{c}_{\tau }\left[\right(|u_1{|}^{\tau -2}{u}_1-|u_2{|}^{\tau -2}{u}_2)(u_1-u_2){]}^{\frac{\tau }{2}}\left(\right|u_1{|}^{\tau }+|u_2{{|}^{\tau })}^{\frac{2-\tau }{\tau }},1<\tau <2,\hfill \\ {\tilde{c}}_{\tau }\left(\right|u_1{|}^{\tau -2}{u}_1-|u_2{|}^{\tau -2}{u}_2)(u_1-u),\tau \ge 2,\hfill \end{array}\right.$$

(43)

for all $u_1,u_2\in {\mathbb{R}}^N$, where $c_{\tau }$ and ${\tilde{c}}_{\tau }$ are positive constants depending only on $\tau$, we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }|\nabla {\eta }_n{-\nabla \eta |}^{\upsilon \left(x\right)}dx\le {\int }_{\Omega }\left(\right|\nabla {\eta }_n{|}^{\upsilon \left(x\right)-2}\nabla {\eta }_n-{|\nabla \eta |}^{\upsilon \left(x\right)-2}\nabla \eta )(\nabla {\eta }_n-\nabla \eta )dx.\end{array}$$

Hence,

$$\parallel {\eta }_n-\eta \parallel \to 0\mathrm{as}n\to \infty ,$$

that is, if ${\eta }_n\rightharpoonup \eta$ in $W_0$ and $\underset{n\to \infty }{lim\; sup}\langle T{\eta }_n-T\eta ,{\eta }_n-\eta \rangle \le 0,$ then ${\eta }_n\to \eta$ in $W_0$. This shows the $\left(S_{+}\right)$-property of T. □

Proof. 

From Section 2, evidently, we know that $W_0$ is a real, separable, and reflexive Banach spaces. Moreover, it follows from Lemmas 6–9 that the operator T satisfies all conditions of the Brezis theorem. Hence, invoking the Brezis theorem, we obtain that $T\eta =0$ has at least one solution $\eta$ in $W_0$, i.e., problem (1) has at least one weak solution $\eta$. □

5. Examples

Now, we give two easy examples of application of our theorems. The first is when $M_{p\left(x\right)}\left(t\right)=a_p+b_{p}t,\mathrm{for}\mathrm{all}t\ge 0\mathrm{with}{a}_p>0,b_p\ge 0$ and $M_{q\left(x\right)}\left(t\right)=a_q+b_{q}t,\mathrm{for}\mathrm{all}t\ge 0\mathrm{with}{a}_q>0,b_q\ge 0.$ In this case, problem (1) reduces to the following form.

Example 1.

Consider the problem

$$\left\{\begin{array}{c}\left(a_p+b_p{\delta }_{p\left(x\right)}\left(\eta \right)\right)(-{\Delta }_{p\left(x\right)}\eta )+\mu \left(a_q+b_q{\delta }_{q\left(x\right)}\left(\eta \right)\right)(-{\Delta }_{q\left(x\right)}\eta )\hfill \\ ={\lambda \left|\eta \right|}^{r\left(x\right)-2}\eta ln\left|\eta \right|+g(x,\eta ,\nabla \eta ),\mathrm{in}\Omega ,\hfill \\ {\eta |}_{\partial \Omega }=0,\hfill \end{array}\right.$$

(44)

where Ω is an open bounded domain in ${\mathbb{R}}^N$ with a smooth boundary.

It is clear that $M_{p\left(x\right)}\left(t\right)\ge a_p>0,\mathrm{for}\mathrm{all}t\ge 0$ and $M_{q\left(x\right)}\left(t\right)\ge a_q>0,\mathrm{for}\mathrm{all}t\ge 0.$ That is, the condition $H_m$ is satisfied. Thus, the results obtained in Theorems 1 and 2 stay true for problem (1). The problem and results are all new.

The second is when $p\left(x\right),q\left(x\right),r\left(x\right),\alpha \left(x\right)$ are constant, that is, $p\left(x\right)=p=constant\in (1,+\infty )$, $q\left(x\right)=q=constant\in (1,+\infty )$, $r\left(x\right)=r=constant\in (1,+\infty )$, $\alpha \left(x\right)=\alpha =constant\in [1,p)$ and $M_{p\left(x\right)}\left(t\right)={(a_p+p{b}_{p}t)}^{p-1},\mathrm{for}\mathrm{all}t\ge 0\mathrm{with}{a}_p>0,b_p\ge 0$ and $M_{q\left(x\right)}\left(t\right)={(a_q+q{b}_{q}t)}^{q-1},\mathrm{for}\mathrm{all}t\ge 0\mathrm{with}{a}_q>0,b_q\ge 0.$ In this case, problem (1) becomes the following form.

Example 2.

Consider the problem

$$\left\{\begin{array}{c}{\left(a_p+b_p{\int }_{\Omega }{|\nabla \eta |}^{p}dx\right)}^{p-1}(-{\Delta }_p\eta )+\mu {\left(a_q+b_q{\int }_{\Omega }{|\nabla \eta |}^{q}dx\right)}^{q-1}(-{\Delta }_q\eta )\hfill \\ ={\lambda \left|\eta \right|}^{r-2}\eta ln\left|\eta \right|+g(x,\eta ,\nabla \eta ),\mathrm{in}\Omega ,\hfill \\ {\eta |}_{\partial \Omega }=0,\hfill \end{array}\right.$$

(45)

where Ω is an open bounded domain in ${\mathbb{R}}^N$ with a smooth boundary.

The function $g:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ given by

$$g(x,\varpi ,\nu )={\left|\varpi \right|}^{\alpha -2}\varpi +\frac{\varpi }{1+{\varpi }^2}\left({\left|\nu \right|}^{p-1}+\gamma \left(x\right)\right),\mathrm{for}\mathrm{all}(x,\varpi ,\nu )\in \Omega \times \mathbb{R}\times {\mathbb{R}}^N,$$

with a constant $\alpha \in [1,p),$ and some $\gamma \in L^{\infty }(\Omega )$ satisfies conditions $H_{g_1}$ (see [21]). For $p\in (1,+\infty ),q\in (1,+\infty )$, the condition $H_{pq}$ is satisfied. It is clear that $M_{p\left(x\right)}\left(t\right)\ge a_p^{p-1}>0,\mathrm{for}\mathrm{all}t\ge 0$ and $M_{q\left(x\right)}\left(t\right)\ge a_q^{q-1}>0,\mathrm{for}\mathrm{all}t\ge 0.$ That is, condition $H_m$ is satisfied. Thus, the results obtained in Theorem 1 stay true for problem (2). The problem and results are also all new.

6. Conclusions

In this article, we study a kind of Kirchhoff-type elliptic problem, which combines with a variable exponent, competing $\left(p\right(x),q(x\left)\right)$-Laplacian, logarithmic nonlinearity, and convection term. Due to the deficit of ellipticity, monotonicity, and variational structure, there are no available techniques to handle problem (1). A fundamental idea of the paper is to seek a solution to (1) as a limit of finite dimensional approximations. With the help of the Galerkin method and Brezis theorem, we obtain the existence of finite-dimensional approximate solutions and weak solutions, respectively. Our study extends previous results, such as from the elliptic problem with logarithmic nonlinearity or the convection term to $\left(p\right(x),q(x\left)\right)$-Kirchhoff-type equations both logarithmic nonlinearity with variable exponents and convection terms. Finally, we consider that it will be a new field to study such problems (1) in fractional Sobolev spaces with variable exponents and in Sobolev spaces with variable exponents and variable fractional order.