Existence of T-ν-p(x)-Solution of a Nonhomogeneous Elliptic Problem with Right Hand Side Measure

El Houcine Rami, Abdelkrim Barbara, El Houssine Azroul

Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco.

DOI: 10.4236/jamp.2021.911175                

Abstract

Using the theory of weighted Sobolev spaces with variable exponent and the L¹-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.

Keywords

Nonhomogeneous Elliptic Equations, Dirichlet Problems, Weighted Sobolev Spaces with Variable Exponent, Minty’s Lemma, T-ν-p(x)-Solutions

1. Introduction

Consider the nonhomogeneous and nonlinear Dirichlet boundary value problem:

$\left(\mathcal{P}\right)\{\begin{array}{ll}-\text{div}\left(a\left(x,u,\nabla u\right)\right)=\mu \hfill & \text{in}\Omega \hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}$

where $\Omega$ is a bounded open domain of $I{R}^N$ ( $N\ge 2$ ) and

$Au=-\text{div}\left(a\left(x\mathrm{,}u\mathrm{,}\nabla u\right)\right)$ is a Leray-Lions operator defined from the weighted Sobolev spaces with variable exponent $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ into its dual $W^{-\mathrm{1,}{p}^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)$ with ${\nu }^{\ast }={\nu }^{1-p^{\prime }\left(x\right)}$ and $\frac{1}{p\left(x\right)}+\frac{1}{p^{\prime }\left(x\right)}=1$. The datum $\mu$ is a measure that admits an L¹-dual composition.

Throughout the paper, we suppose that the exponent $p(\cdot )$ is an element of $C_{+}\left(\bar{\Omega }\right)$ = {log-Hölder continuous function $p(\cdot ):\bar{\Omega }\to IR$ such that $1<p_{-}\le p\left(x\right)\le p_{+}<N$ } (where for all $h\in C_{+}\left(\bar{\Omega }\right)$, we denote $h_{+}$ and $h_{-}$ by $h_{+}={\sup }_{x\in \Omega }h\left(x\right)$ and $h_{-}={\inf }_{x\in \Omega }h\left(x\right)$ ) and that $\nu$ is a weight function defined on $\Omega$ (i.e., $\nu$ is a measurable function which is strictly positive a.e. in $\Omega$ ) satisfying:

$\nu \in L_{loc}^1\left(\Omega \right)\mathrm{,}$ (1.1)

${\nu }^{\frac{-1}{p\left(x\right)-1}}\in L_{loc}^1\left(\Omega \right)\mathrm{,}$ (1.2)

${\nu }^{-s\left(x\right)}\in L^1\left(\Omega \right)\text{for}\text{some}s\left(x\right)\in \left(\frac{N}{p\left(x\right)},\infty \right)\cap \left(\frac{1}{p\left(x\right)-1},\infty \right).$ (1.3)

The problem $\left(\mathcal{P}\right)$ is studied where the following assumptions are satisfied:

(H₁) a is a Carathéodory function satisfying:

$\left|a\left(x\mathrm{,}r\mathrm{,}\xi \right)\right|\le \beta {\nu }^{\frac{1}{p\left(x\right)}}\left[b\left(x\right)+{\left|r\right|}^{p\left(x\right)-1}+{\nu }^{\frac{1}{p^{\prime }\left(x\right)}}{\left(\gamma \left(r\right)\left|\xi \right|\right)}^{p\left(x\right)-1}\right]$ (1.4)

$\left[a\left(x\mathrm{,}r\mathrm{,}\xi \right)-a\left(x\mathrm{,}r\mathrm{,}\eta \right)\right]\left(\xi -\eta \right)\ge 0\forall \xi \mathrm{,}\eta \in I{R}^N$ (1.5)

$a\left(x\mathrm{,}r\mathrm{,}\xi \right)\xi \ge \alpha \nu {\left|\xi \right|}^{p\left(x\right)}\mathrm{,}$ (1.6)

where $b(\cdot )$ is a positive function in $L^{p^{\prime }\left(x\right)}\left(\Omega \right)$, $\gamma \left(r\right)$ is a continuous function and $\alpha \mathrm{,}\beta$ are strictly positive constants.

(H₂) The second member $\mu$ is supposed of the form:

$\mu =f-\text{div}F\mathrm{,}$ (1.7)

where $f\in L^1\left(\Omega \right)$ and $F\in {\left(L^{p^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)\right)}^N$.

A typical example of the problem $\left(\mathcal{P}\right)$ is the following involving the so-called $p\left(x\right)$ -Laplacian operator with weight:

${\Delta }_{\nu \mathrm{,}p\left(x\right)}u=\text{div}\left(\nu \left(x\right){\left|\nabla u\right|}^{p\left(x\right)-2}\nabla u\right)\mathrm{.}$

The operator ${\Delta }_{\nu \mathrm{,}p\left(x\right)}$ becomes p-Laplacian when $p\left(x\right)\equiv p$ (a constant) and $\nu \left(x\right)\equiv 1$. The $p\left(x\right)$ -Laplacian operator with weight possesses more complicated nonlinearities than the classical p-Laplacian, for example, it is inhomogeneous with some degeneracy or singularity. For the applied background of $p\left(x\right)$ -Laplacian, we refer to (see [1] ). The study of differential equations with variable exponents has been a very active field in recent years, we find applications in electro-rheological fluids (see [1] and [2] ) and in image processing (see [3] ).

Under our assumptions (in particular (1.5), the problem $\left(\mathcal{P}\right)$ does not admit, in general, a weak solution since the term $a\left(x\mathrm{,}u\mathrm{,}\nabla u\right)$ may not belong to ${\left(L_{loc}^1\left(\Omega \right)\right)}^N$. To overcome this difficulty we use in this paper the framework of L¹-version of Minty’s lemma (similar to the one used in [4] ). And due to the assumption (1.6) it may be a degenerated or singular problem. Note also that, since the datum is a measure, then the notion of a weak solution cannot be used, hence it is replaced by another approach of solution calling $T\text{-}\nu \text{-}p\left(x\right)$ -solution (see definition 3.1 below).

Dirichlet problem of type $\left(\mathcal{P}\right)$ was considered in ( [5] [6] ), where in the first work the case of $p\left(x\right)\equiv p$ (a constant) and $\nu \left(x\right)\equiv 1$ is treated, while the second work concerns the degenerated case with $p\left(x\right)\equiv p$ (a constant). Hence our present paper can be seen as a generalization of the two works ( [5] [6] ). We also point out that the existence of solutions for elliptic equations with variable exponents can be found in [7] [8] and [9] and.

This paper is divided into three sections, organized as follows: In Section 2, we introduce and prove some properties of the weighted Sobolev spaces with variable exponent and in Section 3, we prove the existence of $T\text{-}\nu \text{-}p\left(x\right)$ -solutions of our problem $\left(\mathcal{P}\right)$. Among the research objectives of this article is to introduce it for applications in physics and also will be a platform for the problem systems of Dirichlet and others.

2. Weighted Sobolev Spaces with Variable Exponent

Let $p\in C_{+}\left(\bar{\Omega }\right)$ and $\nu$ be a weighted function in $\Omega$.

We define the weighted Lebesgue space with variable exponents $L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ as the set of all measurable functions $u\mathrm{<mo>\; :\; </mo>}\Omega \to IR$ for which the convex weight-modular

${\rho }_{\nu \mathrm{,}p\left(x\right)}\left(u\right)={\displaystyle {\int }_{\Omega }}\nu \left(x\right){\left|u\right|}^{p\left(x\right)}\text{d}x$

is finite. The expression

${\Vert u\Vert }_{p\left(x\right),\nu }=\inf \left\{\mu >0:{\displaystyle {\int }_{\Omega }}\nu \left(x\right){\left|\frac{u}{\mu }\right|}^{p\left(x\right)}\text{d}x\le 1\right\}$

defines a norm in $L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, called the Luxemburg norm.

Proposition 2.1. The space $\left(L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right),{\Vert \mathrm{.}\Vert }_{p\left(x\right)\mathrm{,}\nu }\right)$ is a Banach space.

Proof. By considering the operator $M_{{\nu }^{\frac{1}{p\left(x\right)}}}\mathrm{<mo>\; :\; </mo>}{L}^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\to L^{p\left(x\right)}\left(\Omega \right)$ defined by

$M_{{\nu }^{\frac{1}{p\left(x\right)}}}\left(f\right)=f{\nu }^{\frac{1}{p\left(x\right)}}\mathrm{,}$

for all $f\in L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, it’s easy to show that $M_{{\nu }^{\frac{1}{p\left(x\right)}}}$ is an isomorphism and hence we can deduce.

Remark 2.1. When $\nu \left(x\right)\equiv 1$ , the weighted Lebesgue spaces with variable exponent $L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ coincides with the Lebesgue space with variable exponent $L^{p\left(x\right)}\left(\Omega \right)$ .

The weight-modular ${\rho }_{\nu \mathrm{,}p\left(x\right)}$ coincides with the modular ${\rho }_{p\left(x\right)}$ defined on $L^{p\left(x\right)}\left(\Omega \right)$ by ${\rho }_{p\left(x\right)}\left(u\right)\mathrm{<mo>\; :\; </mo>}={\displaystyle {\int }_{\Omega }}{\left|u\right|}^{p\left(x\right)}\text{d}x$ (for more details see [10] [11] [12] and [13] ).

Lemma 2.1. For all function $u\in L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ , the following assertions are satisfied:

1) ${\rho }_{\nu ,p\left(x\right)}\left(u\right)>1\left(=1;<1\right)\iff {\Vert u\Vert }_{p\left(x\right),\nu }>1\left(=1;<1\right)$, respectively.

2) If ${\Vert u\Vert }_{p\left(x\right),\nu }>1$, then ${\Vert u\Vert }_{p\left(x\right)\mathrm{,}\nu }^{p_{-}}\le {\rho }_{\nu \mathrm{,}p\left(x\right)}\left(u\right)\le {\Vert u\Vert }_{p\left(x\right)\mathrm{,}\nu }^{p_{+}}$.

3) If ${\Vert u\Vert }_{p\left(x\right),\nu }<1$, then ${\Vert u\Vert }_{p\left(x\right),\nu }^{p_{+}}\le {\rho }_{\nu ,p\left(x\right)}\left(u\right)\le {\Vert u\Vert }_{p\left(x\right),\nu }^{p_{-}}$.

Proof. It suffices to remark that ${\rho }_{\nu ,p\left(x\right)}\left(u\right)={\rho }_{p\left(x\right)}\left({\nu }^{\frac{1}{p\left(x\right)}}u\right)$ and $\Vert {\nu }^{\frac{1}{p\left(x\right)}}u\Vert ={\Vert u\Vert }_{p\left(x\right),\nu }$, and using the analogous result in [13].

Proposition 2.2. Let $\Omega$ be a bounded open domain of $I{R}^N$ and $\nu$ be a weight function on $\Omega$ satifying the integrability condtions (1.1) and (1.2). Then $L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ ↪ $L_{loc}^1\left(\Omega \right)$.

Proof.

Let K be an included compact on $\Omega$. By vertue of Hölder inequality we have,

$\begin{array}{c}{\displaystyle {\int }_K}\left|u\right|\text{d}x={\displaystyle {\int }_K}\left|u\right|{\nu }^{\frac{1}{p\left(x\right)}}{\nu }^{\frac{-1}{p\left(x\right)}}\text{d}x\\ \le 2{\Vert \left|u\right|{\nu }^{\frac{1}{p\left(x\right)}}\Vert }_{L^{p\left(x\right)}\left(K\right)}{\Vert {\nu }^{\frac{-1}{p\left(x\right)}}\Vert }_{L^{p^{\prime }\left(x\right)}\left(K\right)}\\ \le 2{\Vert u\Vert }_{p\left(x\right),\nu }{\left({\displaystyle {\int }_K}{\nu }^{\frac{-p^{\prime }\left(x\right)}{p\left(x\right)}}\text{d}x+1\right)}^{\frac{1}{{p^{\prime }}_{-}}}\\ \le 2{\Vert u\Vert }_{p\left(x\right),\nu }{\left({\displaystyle {\int }_K}{\nu }^{\frac{-1}{p\left(x\right)-1}}\text{d}x+1\right)}^{\frac{1}{{p^{\prime }}_{-}}}\mathrm{.}\end{array}$

Hence, the conditions (1.1) and (1.2) allow to conclude.

We define the weighted Sobolev space with variable exponents denoted $W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, by

$W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)=\left\{u\in L^{p\left(x\right)}\left(\Omega \right)\mathrm{<mo>\; :\; </mo>}\frac{\partial u}{\partial x_i}\in L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{,}i=\mathrm{1,}\cdots \mathrm{,}N\right\}\mathrm{,}$

equipped with the norm

${\Vert u\Vert }_{\mathrm{1,}p\left(x\right)\mathrm{,}\nu }={\Vert u\Vert }_{p\left(x\right)}+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\Vert \frac{\partial u}{\partial x_i}\Vert }_{p\left(x\right)\mathrm{,}\nu }$

which is equivalent to the Luxemburg norm

$\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}u\mathrm{<mo>\; |\; </mo>\; <mo>\; |\; </mo>\; <mo>\; |\; </mo>}=\inf \left\{\mu >\mathrm{0\; <mo>\; :\; </mo>}{\displaystyle {\int }_{\Omega }}\left({\left|\frac{u}{\mu }\right|}^{p\left(x\right)}+\nu \left(x\right){\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\left|\frac{\frac{\partial u}{\partial x_i}}{\mu }\right|}^{p\left(x\right)}\right)\text{d}x\le 1\right\}\mathrm{.}$

Proposition 2.3. Let $\nu$ be a weight function on $\Omega$ satisfying the conditions (1.1) and (1.2). Then the space $\left(W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{,}{\Vert \mathrm{.}\Vert }_{\mathrm{1,}p\left(x\right)\mathrm{,}\nu }\right)$ is a Banach space.

Proof. Let ${\left(u_n\right)}_n$ be a Cauchy sequence in $\left(W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{,}{\Vert \mathrm{.}\Vert }_{\mathrm{1,}p\left(x\right)\mathrm{,}\nu }\right)$. Then ${\left(u_n\right)}_n$ is a Cauchy sequence in $L^{p\left(x\right)}\left(\Omega \right)$ and ${\left(\frac{\partial u_n}{\partial x_i}\right)}_n$ is also a Cauchy sequence in $L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ for all $i=1,\cdots ,N$. By vertue of proposition 2.1, we can deduce that there exist $u\in L^{p\left(x\right)}\left(\Omega \right)$ and $v_i\in L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ such that:

$u_n\to u\text{in}{L}^{p\left(x\right)}(\; \Omega \; )$

and

$\frac{\partial u_n}{\partial x_i}\to v_i\text{in}{L}^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\text{for}\text{all}i=\mathrm{1,}\cdots \mathrm{,}N\mathrm{.}$

Moreover, by using proposition 2.2, we have $L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\subset L_{loc}^1\left(\Omega \right)\subset D^{\prime }\left(\Omega \right)$. Thus, for all $\phi \in D\left(\Omega \right)$ one has,

$\langle T_{v_i},\phi \rangle ={\lim }_{n\to \infty }\langle T_{\frac{\partial u_n}{\partial x_i}},\phi \rangle =-{\lim }_{n\to \infty }\langle T_{u_n},\frac{\partial \phi }{\partial x_i}\rangle =-\langle T_u,\frac{\partial \phi }{\partial x_i}\rangle =\langle T_{\frac{\partial u}{\partial x_i}},\phi \rangle .$

Hence $T_{v_i}=T_{\frac{\partial u}{\partial x_i}}$, i.e. $v_i=\frac{\partial u}{\partial x_i}$.

Consequently,

$u\in W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$

and

$u_n\to u\text{in}{W}^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{.}$

Remark 2.2. Since $\nu$ satisfies the conditions (1.1) and (1.2), it’s easy to prove that $C_0^{\infty }\left(\Omega \right)$ is included in $W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ ; then we can define the following space

$W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)={\overline{C_0^{\infty }\left(\Omega \right)}}^{{\Vert \mathrm{.}\Vert }_{\mathrm{1,}p\left(x\right)\mathrm{,}\nu }}\mathrm{,}$

which is also a Banach space under the norm ${\Vert \mathrm{.}\Vert }_{\mathrm{1,}p\left(x\right)\mathrm{,}\nu }$.

Proposition 2.4. (Characterization of the dual space).

Let $p\left(\mathrm{.}\right)\in C_{+}\left(\bar{\Omega }\right)$ and $\nu$ be a weight function on $\Omega$ satisfying the conditions (1.1) and (1.2). Then for all $G\in {\left(W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\right)}^{\ast }$, there exists a unique system of functions $\left(g_0\mathrm{,}{g}_1\mathrm{,}\cdots \mathrm{,}{g}_N\right)\in L^{p^{\prime }\left(x\right)}\left(\Omega \right)\times {\left(L^{p^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{1-p^{\prime }\left(x\right)}\right)\right)}^N$ such that,

$G\left(f\right)={\displaystyle {\int }_{\Omega }}f\left(x\right)g_0\left(x\right)\text{d}x+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}\frac{\partial f}{\partial x_i}{g}_i\left(x\right)\text{d}x\mathrm{,}\forall f\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{.}$

Proof. The proof of this proposition is similar to that used in [12] (theorem3.16).

Now, let us introduce the function $p_s$ defined by

$p_s\left(x\right)=\frac{p\left(x\right)s\left(x\right)}{s\left(x\right)+1}\mathrm{.}$

We have

$p_s\left(x\right)<p\left(x\right)\text{a}\text{.e}\text{.}\text{in}\Omega$

and

$\{\begin{array}{l}{p}_s^{\ast }\left(x\right)=\frac{N{p}_s\left(x\right)}{N-p_s\left(x\right)}=\frac{Np\left(x\right)s\left(x\right)}{N\left(s\left(x\right)+1\right)-p\left(x\right)s\left(x\right)}\text{if}p\left(x\right)s\left(x\right)<N\left(s\left(x\right)+1\right),\\ p_s^{\ast }\left(x\right)\text{is}\text{arbitrary}\mathrm{,}\text{otherwise}\mathrm{.}\end{array}$

Proposition 2.5. Let $p\mathrm{,}s\in C_{+}\left(\bar{\Omega }\right)$ and $\nu$ be a weight function on $\Omega$ which satisfies the conditions (1.1), (1.2) and (1.3). Then $W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ ↪ $W^{\mathrm{1,}{p}_s\left(x\right)}\left(\Omega \right)$.

Proof. According to the Hölder inequality and the condition (1.3), one has

$\begin{array}{c}{\displaystyle {\int }_{\Omega }}{\left|v\left(x\right)\right|}^{p_s\left(x\right)}\text{d}x={\displaystyle {\int }_{\Omega }}{\left|v\left(x\right)\right|}^{p_s\left(x\right)}{\nu }^{\frac{p_s\left(x\right)}{p\left(x\right)}}{\nu }^{\frac{-p_s\left(x\right)}{p\left(x\right)}}\text{d}x\\ \le \left(\frac{1}{{\left(\frac{p}{p_s}\right)}_{-}}+\frac{1}{{\left(s+1\right)}^{-}}\right){\Vert {\left|v\left(x\right)\right|}^{p_s\left(x\right)}{\nu }^{\frac{p_s\left(x\right)}{p\left(x\right)}}\Vert }_{\frac{p\left(x\right)}{p_s\left(x\right)}}{\Vert {\nu }^{\frac{-p_s\left(x\right)}{p\left(x\right)}}\Vert }_{s\left(x\right)+1}\\ \le \left(\frac{1}{{\left(\frac{p}{p_s}\right)}_{-}}+\frac{1}{{\left(s+1\right)}^{-}}\right){\left({\displaystyle {\int }_{\Omega }}{\left|v\left(x\right)\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\right)}^{\frac{1}{{\gamma }_1}}{\left({\displaystyle {\int }_{\Omega }}\nu {\left(x\right)}^{-s\left(x\right)}\text{d}x\right)}^{\frac{1}{\overline{{\gamma }_1}}}\\ \le C{\left({\displaystyle {\int }_{\Omega }}{\left|v\left(x\right)\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\right)}^{\frac{1}{{\gamma }_1}}{\left({\displaystyle {\int }_{\Omega }}\nu {\left(x\right)}^{-s\left(x\right)}\text{d}x\right)}^{\frac{1}{\overline{{\gamma }_1}}}\\ \le C{\left({\displaystyle {\int }_{\Omega }}{\left|v\left(x\right)\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\right)}^{\frac{1}{{\gamma }_1}}\mathrm{.}\end{array}$

If we take $v=\frac{\partial u}{\partial x_i}$, we then obtain

${\displaystyle {\int }_{\Omega }}{\left|\frac{\partial u}{\partial x_i}\right|}^{p_s\left(x\right)}\text{d}x\le C{\left({\displaystyle {\int }_{\Omega }}{\left|\frac{\partial u}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\right)}^{\frac{1}{{\gamma }_1}}$

where

${\gamma }_1=\{\begin{array}{ll}{\left(\frac{p}{p_s}\right)}_{-}\hfill & \text{if}{\Vert {\left|\frac{\partial u}{\partial x_i}\left(x\right)\right|}^{p_s\left(x\right)}{\nu }^{\frac{p_s\left(x\right)}{p\left(x\right)}}\Vert }_{\frac{p\left(x\right)}{p_s\left(x\right)}}\ge 1,\hfill \\ {\left(\frac{p}{p_s}\right)}^{+}\hfill & \text{if}{\Vert {\left|\frac{\partial u}{\partial x_i}\left(x\right)\right|}^{p_s\left(x\right)}{\nu }^{\frac{p_s\left(x\right)}{p\left(x\right)}}\Vert }_{\frac{p\left(x\right)}{p_s\left(x\right)}}<1.\hfill \end{array}$

Consequently, we can write

${\Vert \frac{\partial u}{\partial x_i}\left(x\right)\Vert }_{p_s\left(x\right)}^{{\gamma }_2}\le C{\left({\displaystyle {\int }_{\Omega }}{\left|\frac{\partial u}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\right)}^{\frac{1}{{\gamma }_1}}\le C_0{C}_1{\Vert \frac{\partial u}{\partial x_i}\left(x\right)\Vert }_{p\left(x\right)\mathrm{,}\nu }^{\frac{{\gamma }_3}{{\gamma }_1}}$

where

${\gamma }_2=\{\begin{array}{ll}{\left(p_s\right)}_{-}\hfill & \text{if}{\Vert \frac{\partial u}{\partial x_i}\left(x\right)\Vert }_{p_s\left(x\right)}\ge 1,\hfill \\ {\left(p_s\right)}^{+}\hfill & \text{if}{\Vert \frac{\partial u}{\partial x_i}\left(x\right)\Vert }_{p_s\left(x\right)}<1,\hfill \end{array}$

and

${\gamma }_3=\{\begin{array}{ll}{p}^{+}\hfill & \text{si}{\Vert \frac{\partial u}{\partial x_i}\left(x\right)\Vert }_{p\left(x\right),\nu }\ge 1,\hfill \\ p_{-}\hfill & \text{si}{\Vert \frac{\partial u}{\partial x_i}\left(x\right)\Vert }_{p\left(x\right),\nu }<1.\hfill \end{array}$

Thus

${\Vert \frac{\partial u}{\partial x_i}\Vert }_{p_s\left(x\right)}\le C{\Vert \frac{\partial u}{\partial x_i}\Vert }_{p\left(x\right)\mathrm{,}\nu }^{\frac{{\gamma }_3}{{\gamma }_1{\gamma }_2}}\mathrm{,}i=\mathrm{1,2,}\cdots \mathrm{,}N\mathrm{.}$ (2.1)

Note that $C=c\left({\gamma }_1\mathrm{,}{\gamma }_2\mathrm{,}{\gamma }_3\right)$ denotes some positive constant which may be changing step by step.

Since $p_s\left(x\right)<p\left(x\right)$ p.p. in $\Omega$, then, there exists a positive constant C such that

${\Vert u\Vert }_{L^{p_s\left(x\right)}\left(\Omega \right)}\le C{\Vert u\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\mathrm{.}$

Thus, we conclude that

$W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ ↪ $W^{\mathrm{1,}{p}_s\left(x\right)}\left(\Omega \right)$.

Corollary 2.1. Let $p\mathrm{,}s\in C_{+}\left(\bar{\Omega }\right)$ and $\nu$ be a weight on $\Omega$ which satisfies the conditions (1.1), (1.2) and (1.3). Then $W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ ↪↪ $L^{r\left(x\right)}\left(\Omega \right)$, for $1\le r\left(x\right)<p_s^{\ast }\left(x\right)$ .

Corollary 2.2. Let $p\in C_{+}\left(\bar{\Omega }\right)$ and $\nu$ be a weight function on $\Omega$ which satisfies the conditions (1.1), (1.2) and (1.3). Then

${\Vert u\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\le C{\Vert \nabla u\Vert }_{L^{p\left(x\right)}\left(\Omega \mathrm{<mo>\; ;\; </mo>}\nu \right)}\mathrm{,}\forall u\in {\mathcal{C}}_0^{\infty }\left(\Omega \right)\mathrm{.}$

Proof. Let $u\in {\mathcal{C}}_0^{\infty }\left(\Omega \right)$. Since $1\le p\left(x\right)<p_s^{\ast }\left(x\right)$, we deduce by vertue of the embedding $W^{\mathrm{1,}{p}_s\left(x\right)}\left(\Omega \right)$ ↪ $L^{p\left(x\right)}\left(\Omega \right)$ that,

${\Vert u\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\le C_1\left({\Vert u\Vert }_{L^{p_s\left(x\right)}\left(\Omega \right)}+{\Vert \nabla u\Vert }_{{\left(L^{p_s\left(x\right)}\left(\Omega \right)\right)}^N}\right)\mathrm{.}$

Thus, in view of the proposition 2.5, we obtain

${\Vert u\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\le C_2{\Vert \nabla u\Vert }_{L^{p_s}\left(\Omega \right)}\le C_3{\Vert \nabla u\Vert }_{L^{p\left(x\right)}\left(\Omega \mathrm{<mo>\; ;\; </mo>}\nu \right)}\mathrm{,}$

which allows to conclude that

${\Vert u\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\le C{\Vert \nabla u\Vert }_{L^{p\left(x\right)}\left(\Omega \mathrm{<mo>\; ;\; </mo>}\nu \right)}\mathrm{.}$

3. Existence Result

Consider the nonhomogeneous nonlinear Dirichlet boundary problem:

$\left(\mathcal{P}\right)\{\begin{array}{l}-\text{div}\left(a\left(x\mathrm{,}u\mathrm{,}\nabla u\right)\right)=-\text{div}F\text{in}\Omega \\ u=0\text{on}\partial \Omega \mathrm{.}\end{array}$

Definition 3.1. A function u is called a $T\text{-}\nu \text{-}p\left(x\right)$ -solution of problem $\left(\mathcal{P}\right)$ if:

$\{\begin{array}{l}u\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{,}\\ {\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}u\mathrm{,}\nabla u\right)\nabla T_k\left(u-\phi \right)\text{d}x={\displaystyle {\int }_{\Omega }}f{T}_k\left(u-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u-\phi \right)\text{d}x\mathrm{,}\forall \phi \in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\cap L^{\infty }\left(\Omega \right)\mathrm{.}\end{array}$

Theorem 3.1. Let suppose that the assumptions (1.1)-(1.7) are satisfied. Then the problem $\left(\mathcal{P}\right)$ has at least one $T\text{-}\nu \text{-}p\left(x\right)$ -solution.

Remark 3.1. Note that in the particular case where $p\left(\mathrm{.}\right)\equiv p$ (constant), $\gamma \left(r\right)=1$ and $\nu =1$ , the same result is proved in [14] by using the approach of pseudo-monotonicity.

3.1. Approximate Problem

Let ${\left(f_n\right)}_n$ be a sequence of functions in $L^{\infty }\left(\Omega \right)$ which converges strongly to f in $L^1\left(\Omega \right)$ such that ${\Vert f_n\Vert }_{L^{\infty }\left(\Omega \right)}\le {\Vert f\Vert }_{L^{\infty }\left(\Omega \right)}$. For $n\ge 1$, we consider the approximate problem of $(\; P\; )$

$\left({\mathcal{P}}_n\right)\{\begin{array}{l}{u}_n\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\\ -\text{div}\left(a\left(x\mathrm{,}{T}_n\left(u_n\right)\mathrm{,}\nabla u_n\right)\right)=f_n-\text{div}F\text{in}\Omega \mathrm{.}\end{array}$

This section is devoted to establishing the existing solution for the approximate problem $\left({\mathcal{P}}_n\right)$.

Theorem 3.2. The operator $A_k$ defined by,

$\begin{array}{l}{A}_k\mathrm{<mo>\; :\; </mo>}{W}_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\to W^{-\mathrm{1,}{p}^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)\\ u\mapsto A_{k}u=-\text{div}\left(a\left(x\mathrm{,}{T}_k\left(u\right)\mathrm{,}\nabla u\right)\right)\end{array}$

is bounded, coercive, hemicontinuous and pseudo-monotone.

Proof of Theorem 3.2

● The operator $A_k$ is bounded. Indeed for all $u\mathrm{,}v\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, one has

$\begin{array}{l}\left|\langle A_{k}u\mathrm{,}v\rangle \right|=\left|{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{T}_k\left(u\right)\mathrm{,}\nabla u\right)\nabla v\text{d}x\right|=\left|{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{T}_k\left(u\right)\mathrm{,}\nabla u\right){\nu }^{\frac{-1}{p\left(x\right)}}\nabla v{\nu }^{\frac{1}{p\left(x\right)}}\text{d}x\right|\\ \le \left(\frac{1}{p_{-}}+\frac{1}{{p^{\prime }}_{-}}\right){\Vert a\left(x\mathrm{,}{T}_k\left(u\right)\mathrm{,}\nabla u\right){\nu }^{\frac{-1}{p\left(x\right)}}\Vert }_{L^{p^{\prime }\left(x\right)}\left(\Omega \right)}{\Vert \nabla v{\nu }^{\frac{1}{p\left(x\right)}}\Vert }_{L^{p\left(x\right)}\left(\Omega \right)}\\ \le 2{\left({\displaystyle {\int }_{\Omega }}{\left|a\left(x\mathrm{,}{T}_k\left(u\right)\mathrm{,}\nabla u\right){\nu }^{\frac{-1}{p\left(x\right)}}\right|}^{p^{\prime }\left(x\right)}\text{d}x\right)}^{\frac{1}{{p^{\prime }}_{-}}}{\Vert \nabla v\Vert }_{L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}\\ \le 2{\left({\displaystyle {\int }_{\Omega }}{\left(b\left(x\right)+{\left|T_k\left(u\right)\right|}^{p\left(x\right)-1}+{\nu }^{\frac{1}{p^{\prime }\left(x\right)}}{\left(\gamma \left(T_k\left(u\right)\right)\left|\nabla u\right|\right)}^{p\left(x\right)-1}\right)}^{p^{\prime }\left(x\right)}\text{d}x\right)}^{\frac{1}{{p^{\prime }}_{-}}}{\Vert \nabla v\Vert }_{L^{p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}\\ \le C_1{\left({\displaystyle {\int }_{\Omega }}\left(b{\left(x\right)}^{p^{\prime }\left(x\right)}+{\left|T_k\left(u\right)\right|}^{p\left(x\right)}+\nu \left(x\right){\left(\gamma \left(T_k\left(u\right)\right)\left|\nabla u\right|\right)}^{p\left(x\right)}\right)\text{d}x\right)}^{\frac{1}{{p^{\prime }}_{-}}}{\Vert v\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}\end{array}$

$\le \left(C_1+C_2+C_3{\left({\displaystyle {\int }_{\Omega }}{\left|T_k\left(u\right)\right|}^{p\left(x\right)}+\nu \left(x\right){\left(\gamma \left(T_k\left(u\right)\right)\left|\nabla u\right|\right)}^{p\left(x\right)}\text{d}x\right)}^{\frac{1}{{p^{\prime }}_{-}}}\right){\Vert v\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}\mathrm{.}$

Since $\gamma \left(\mathrm{.}\right)$ is continuous and $\left|T_k\left(u\right)\right|\le k$ a.e. in $\Omega$, then $\gamma \left(T_k\left(u\right)\right)\left|\nabla u\right|$ is bounded in $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ ; hence the operator $A_k$ is bounded.

● The operator $A_k$ is hemicontinuous. Indeed, let t be a reality that tends to $t_0$. We have

$a\left(x\mathrm{,}{T}_k\left(u+tv\right)\mathrm{,}\nabla T_k\left(u+tv\right)\right)\to a\left(x\mathrm{,}{T}_k\left(u+t_{0}v\right)\mathrm{,}\nabla T_k\left(u+t_{0}v\right)\right)\mathrm{,}\text{a}\text{.e}\text{.}\text{in}\Omega \mathrm{.}$

Since ${\left(a\left(x\mathrm{,}{T}_k\left(u+tv\right)\mathrm{,}\nabla T_k\left(u+tv\right)\right)\right)}_t$ is bounded in ${\left(L^{p^{\prime }}\left(\Omega \right)\right)}^N$, we deduce that $A_k\left(u+tv\right)$ converges to $A_k\left(u+t_{0}v\right)$ weakly in $W^{-\mathrm{1,}{p}^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)$ as t tends to $t_0$.

● The operator $A_k$ is coercive. Indeed, for all $u\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, we have

$\frac{\langle A_{k}u\mathrm{,}u\rangle }{{\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}}\ge \frac{{\displaystyle {\int }_{\Omega }}\nu \left(x\right){\left|\nabla u\right|}^{p\left(x\right)}\text{d}x}{{\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}}\ge \frac{{\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}^{\delta }}{{\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}}\ge {\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}^{\delta -1}\mathrm{,}$

where

$\delta =\{\begin{array}{ll}{p}_{-}\hfill & \text{if}{\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}\le \mathrm{1,}\hfill \\ p^{+}\hfill & \text{if}{\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}>\mathrm{1,}\hfill \end{array}$

Obviously, we have ${\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}^{\delta -1}$ tends to infinity, when ${\Vert u\Vert }_{W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)}\to \infty$, hence we conclude.

● It remains to show that $A_k$ is pseudo-monotone: Let ${\left(u_j\right)}_j$ be a sequence in $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ such that

$u_j\rightharpoonup u\text{in}{W}_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\text{and}\underset{j}{\lim \sup }\langle A_k{u}_j\mathrm{,}{u}_j-u\rangle \le 0.$ (3.1)

Firstly, we prove that $A_k{u}_j$ converges to $A_{k}u$ weakly in $W^{-\mathrm{1,}{p}^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)$. Indeed, since ${\left(u_j\right)}_j$ is a bounded sequence in $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, then by the growth condition, ${\left(A_k{u}_j\right)}_j$ is bounded in $W^{-\mathrm{1,}{p}^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)$, therefore there exists a function $h_k=\left(h_{ki}\right)$ such that,

$\begin{array}{l}{A}_k{u}_j\rightharpoonup h_k\text{dans}{W}^{-\mathrm{1,}{p}^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)\mathrm{,}\\ a_i\left(x\mathrm{,}{T}_k\left(u_j\right)\mathrm{,}\nabla u_j\right)\rightharpoonup h_{ki}\text{in}{L}^{p^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)\mathrm{,}\text{for}i=\mathrm{1,}\cdots \mathrm{,}N\mathrm{.}\end{array}$ (3.2)

Hence, we can write

$\underset{j}{\lim \sup }\langle A_k{u}_j\mathrm{,}{u}_j\rangle \le \langle h_k\mathrm{,}u\rangle \mathrm{.}$ (3.3)

On the one hand, by (1.5), we have

$\begin{array}{l}{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}\left(a_i\left(x\mathrm{,}{T}_k\left(u_j\right)\mathrm{,}\nabla v\right)-a_i\left(x\mathrm{,}{T}_k\left(u_j\right)\mathrm{,}\nabla u_j\right)\right)\left(\frac{\partial v}{\partial x_i}-\frac{\partial u_j}{\partial x_i}\right)\text{d}x\ge \mathrm{0,}\\ \forall v\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{.}\end{array}$

Then

$\begin{array}{l}{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial u_j}{\partial x_i}dx\\ \ge {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial v}{\partial x_i}\text{d}x-{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla v\right)\frac{\partial v}{\partial x_i}\text{d}x\\ +{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla v\right)\frac{\partial u_j}{\partial x_i}\text{d}x.\end{array}$ (3.4)

Since $u_j\to u$ strongly in $L^{p\left(x\right)}\left(\Omega \right)$ and a.e. in $\Omega$, then

$a_i\left(x\mathrm{,}{T}_k\left(u_j\right)\mathrm{,}\nabla v\right)\to a_i\left(x\mathrm{,}{T}_k\left(u\right)\mathrm{,}\nabla v\right)\text{strongly}\text{in}{L}^{p^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)\text{for}i=\mathrm{1,}\cdots \mathrm{,}N\mathrm{.}$ (3.5)

Therefore,

${\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla v\right)\frac{\partial v}{\partial x_i}\text{d}x\to {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u\right),\nabla v\right)\frac{\partial v}{\partial x_i}\text{d}x$ (3.6)

and

${\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla v\right)\frac{\partial u_j}{\partial x_i}\text{d}x\to {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u\right),\nabla v\right)\frac{\partial u}{\partial x_i}\text{d}x.$ (3.7)

By vertue of (3.2), we have

${\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial v}{\partial x_i}\text{d}x\to {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{h}_{ki}\frac{\partial v}{\partial x_i}\text{d}x.$ (3.8)

Now, combining (3.4)-(3.6) and (3.7), we obtain

$\begin{array}{l}\underset{j\to \infty }{\lim }{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial u_j}{\partial x_i}\text{d}x\\ \ge {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{h}_{ki}\frac{\partial v}{\partial x_i}\text{d}x+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u\right),\nabla v\right)\frac{\partial u}{\partial x_i}\text{d}x\\ -{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u\right),\nabla v\right)\frac{\partial v}{\partial x_i}\text{d}x.\end{array}$

Due to (3.3), we deduce that

$\begin{array}{c}{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{h}_{ki}\frac{\partial u}{\partial x_i}\text{d}x\ge {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{h}_{ki}\frac{\partial v}{\partial x_i}\text{d}x+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u\right),\nabla v\right)\frac{\partial u}{\partial x_i}\text{d}x\\ -{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u\right),\nabla v\right)\frac{\partial v}{\partial x_i}\text{d}x.\end{array}$

This implies that,

${\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}\left(a_i\left(x,T_k\left(u_j\right),\nabla v\right)-h_{ki}\right)\left(\frac{\partial v}{\partial x_i}-\frac{\partial u_j}{\partial x_i}\right)\text{d}x\ge 0,\forall v\in W_0^{1,p\left(x\right)}\left(\Omega ,\nu \right).$ (3.9)

On the other hand, choose $v=u+tw$ in (3.9) (with $t\in \left]-\mathrm{1,1}\right[$ ). It’s easy to see that

${\displaystyle {\int }_{\Omega }}\left(a\left(x\mathrm{,}{T}_k\left(u\right)\mathrm{,}\nabla \left(u+tw\right)\right)-h_k\right)\nabla w\text{d}x=\mathrm{0,}\forall w\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{,}\forall t\in \left]-\mathrm{1,1}\right[\mathrm{.}$

Hence $A_{k}u=h_k\in W^{-1,p^{\prime }\left(x\right)}\left(\Omega ,{\nu }^{\ast }\right)$, and we deduce that $A_k{u}_j$ weakly converges to $A_{k}u$ in $W^{-1,p^{\prime }\left(x\right)}\left(\Omega ,{\nu }^{\ast }\right)$.

Secondly, we prove that $\langle A_k{u}_j\mathrm{,}{u}_j\rangle \to \langle A_{k}u\mathrm{,}u\rangle$. Indeed, in view of (3.2) and (3.3), we have

$\lim \sup \langle A_k{u}_j,u_j\rangle \le \langle A_{k}u,u\rangle =\langle h_k,u\rangle .$

It remains to show that,

$\lim \inf \langle A_k{u}_j,u_j\rangle \ge \langle A_{k}u,u\rangle =\langle h_k,u\rangle .$

For that, we have

$\begin{array}{c}\langle A_k{u}_j,u_j\rangle ={\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial u_j}{\partial x_i}\text{d}x\\ ={\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}\left(a_i\left(x,T_k\left(u_j\right),\nabla u_j\right)-a_i\left(x,T_k\left(u_j\right),\nabla u\right)\right)\left(\frac{\partial u_j}{\partial x_i}-\frac{\partial u}{\partial x_i}\right)\text{d}x\\ +{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u\right)\left(\frac{\partial u_j}{\partial x_i}-\frac{\partial u}{\partial x_i}\right)\text{d}x\\ +{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial u}{\partial x_i}\text{d}x.\end{array}$

Since ${\displaystyle {\sum }_{i=1}^N}{\displaystyle {\int }_{\Omega }}\left(a_i\left(x,T_k\left(u_j\right),\nabla u_j\right)-a_i\left(x,T_k\left(u_j\right),\nabla u\right)\right)\left(\frac{\partial u_j}{\partial x_i}-\frac{\partial u}{\partial x_i}\right)\text{d}x\ge 0$, we deduce that

$\begin{array}{c}\langle A_k{u}_j,u_j\rangle \ge {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u\right))\left(\frac{\partial u_j}{\partial x_i}-\frac{\partial u}{\partial x_i}\right)\text{d}x\\ +{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial u}{\partial x_i}\text{d}x.\end{array}$

Therefore,

$\begin{array}{c}\lim \inf \langle A_k{u}_j,u_j\rangle \ge \lim \inf {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u\right)\left(\frac{\partial u_j}{\partial x_i}-\frac{\partial u}{\partial x_i}\right)\text{d}x\\ +\lim \inf {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{a}_i\left(x,T_k\left(u_j\right),\nabla u_j\right)\frac{\partial u}{\partial x_i}\text{d}x.\end{array}$

Hence, $\lim \inf \langle A_k{u}_j,u_j\rangle \ge {\displaystyle {\sum }_{i=1}^N}{\displaystyle {\int }_{\Omega }}{h}_i\frac{\partial u}{\partial x_i}\text{d}x\ge \langle A_{k}u,u\rangle$. This achieved the proof.

3.2. Proof of Theorem 3.1

The proof is divided into 4 steps.

Step 1: We will show that ${\left(u_n\right)}_n$ is a Cauchy sequence in measure. Using $T_k\left(u_n\right)$ as a test function in $\left({\mathcal{P}}_n\right)$ leads to,

${\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{T}_k\left(u_n\right)\mathrm{,}\nabla u_n\right)\nabla T_k\left(u_n\right)\text{d}x={\displaystyle {\int }_{\Omega }}{f}_n{T}_k\left(u_n\right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\cdot \nabla T_k\left(u_n\right)\text{d}x\mathrm{.}$

From (1.6) and (1.7), we deduce for all $k>1$ that,

$\begin{array}{l}\alpha {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\\ \le k{\Vert f\Vert }_{L^1}+{\displaystyle \underset{i\mathrm{<mo>\; =\; </mo>1}}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}\left|F_i\right|\nu {\left(x\right)}^{\frac{-1}{p\left(x\right)}}\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|\nu {\left(x\right)}^{\frac{1}{p\left(x\right)}}\text{d}x\\ \le k{\Vert f\Vert }_{L^1}+{\displaystyle \underset{i\mathrm{<mo>\; =\; </mo>1}}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}\left|F_i\right|\nu {\left(x\right)}^{\frac{-1}{p\left(x\right)}}{\left(\frac{\alpha }{2}\right)}^{\frac{-1}{p\left(x\right)}}\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|\nu {\left(x\right)}^{\frac{1}{p\left(x\right)}}{\left(\frac{\alpha }{2}\right)}^{\frac{1}{p\left(x\right)}}\text{d}x\mathrm{.}\end{array}$

Now, by Young’s inequality, we obtain

$\begin{array}{l}\alpha {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\\ \le k{\Vert f\Vert }_{L^1}+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|F_i\right|}^{p^{\prime }\left(x\right)}\nu {\left(x\right)}^{\frac{-p^{\prime }\left(x\right)}{p\left(x\right)}}\frac{C\left(\alpha \right)}{p^{\prime }\left(x\right)}\text{d}x+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\frac{\alpha }{2p\left(x\right)}\text{d}x\\ \le k{\Vert f\Vert }_{L^1}+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|F_i\right|}^{p^{\prime }\left(x\right)}\nu {\left(x\right)}^{\frac{-p^{\prime }\left(x\right)}{p\left(x\right)}}C\left(\alpha ,{p^{\prime }}^{-}\right)\text{d}x+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\frac{\alpha }{2p^{-}}\text{d}x.\end{array}$ (3.10)

Then, one has

$\begin{array}{l}\left(1-\frac{1}{2p^{-}}\right)\alpha {\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\\ \le k{\Vert f\Vert }_{L^1}+\frac{C\left(\alpha \mathrm{,}{p^{\prime }}^{-}\right)}{k}+{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|F_i\right|}^{p^{\prime }\left(x\right)}\nu {\left(x\right)}^{\frac{-p^{\prime }\left(x\right)}{p\left(x\right)}}\text{d}x\mathrm{,}\end{array}$

for $k\ge 1$, which implies that

${\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\le Ck\text{for}\text{all}k>1.$ (3.11)

Let $k>0$ large enough and $B_R$ be a ball of $\Omega$. Using (3.11) and applying Hölder’s inequality and Poincaré’s inequality, we obtain

$\begin{array}{l}kmeas\left(\left\{\left|u_n\right|>k\right\}\cap B_R\right)\\ ={\displaystyle {\int }_{\left\{\left|u_n\right|>k\right\}\cap B_R}}\left|T_k\left(u_n\right)\right|\text{d}x\le {\Vert T_k\left(u_n\right)\Vert }_{L^1\left(\Omega \right)}\le C{\Vert T_k\left(u_n\right)\Vert }_{L^{p\left(x\right)}(\; \Omega \; )}\end{array}$

$\le C{\Vert \nabla T_k\left(u_n\right)\Vert }_{p\left(x\right),\nu }$ (by vertue of Corollary 2.2) (3.12)

$\le C{\left({\displaystyle {\int }_{\Omega }}{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\left|\frac{\partial T_k\left(u_n\right)}{\partial x_i}\right|}^{p\left(x\right)}\nu \left(x\right)\text{d}x\right)}^{\frac{1}{\kappa }}$ (by vertue of Lemma 2.1)

$\le C{k}^{\frac{1}{\kappa }},$

where

$\kappa =\{\begin{array}{ll}{p}_{-}\hfill & \text{if}{\Vert \nabla T_k\left(u_n\right)\Vert }_{p\left(x\right),\nu }\le 1,\hfill \\ p_{+}\hfill & \text{if}{\Vert \nabla T_k\left(u_n\right)\Vert }_{p\left(x\right),\nu }>1,\hfill \end{array}$

which implies that,

$meas\left(\left\{\left|u_n\right|>k\right\}\cap B_R\right)\le \frac{C}{k^{1-\frac{1}{\kappa }}}\mathrm{,}\forall k>1.$ (3.13)

So, we have, for all $\delta >0$,

$\begin{array}{l}meas\left(\left\{\left|u_n-u_m\right|>\delta \right\}\cap B_R\right)\\ \le meas\left(\left\{\left|u_n\right|>k\right\}\cap B_R\right)+meas\left(\left\{\left|u_m\right|>k\right\}\cap B_R\right)\\ +meas\left(\left\{\left|T_k\left(u_n\right)-T_k\left(u_m\right)\right|>\delta \right\}\right)\mathrm{.}\end{array}$ (3.14)

Since ${\left(T_k\left(u_n\right)\right)}_n$ is bounded in $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, there exists a subsequence, still denoted by $T_k\left(u_n\right)$ and a measurable function $v_k\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ such that $T_k\left(u_n\right)$ converges to $v_k$ weakly in $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, strongly in $L^{p\left(x\right)}\left(\Omega \right)$ and

almost everywhere in $\Omega$. Hence ${\left(T_k\left(u_n\right)\right)}_n$ is a Cauchy sequence in measure in $\Omega$.

Let $\epsilon >0$. Then by (3.13), there exists $k\left(\epsilon \right)>0$ such that,

$meas\left(\left\{\left|u_n-u_m\right|>\delta \right\}\cap B_R\right)<\epsilon ,\forall n,m\ge n_0\left(k\left(\epsilon \right),\delta ,R\right).$

This proves that ${\left(u_n\right)}_n$ is a Cauchy sequence in measure in $B_R$, thus converges almost everywhere to some measurable function u. Hence

$\begin{array}{l}{T}_k\left(u_n\right)\rightharpoonup T_k\left(u\right)\text{weakly}\text{in}{W}_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{,}\\ \text{strongly}\text{in}{W}^{p\left(x\right)}\left(\Omega \right)\mathrm{,}\text{and}\text{a}\text{.e}\text{.}\text{in}\Omega \mathrm{.}\end{array}$ (3.15)

Step 2: We shall prove that

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla \phi \right)\nabla T_k\left(u_n-\phi \right)\text{d}x\\ \le {\displaystyle {\int }_{\Omega }}{f}_n{T}_k\left(u_n-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u_n-\phi \right)\text{d}x\\ \forall \phi \in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\cap L^{\infty }\left(\Omega \right)\mathrm{.}\end{array}$ (3.16)

Let $\phi \in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\cap L^{\infty }\left(\Omega \right)$ and let n be large enough ( $n\ge k+{\Vert \phi \Vert }_{\infty }$ ). Using the admissible test function $T_k\left(u_n-\phi \right)$ in $\left({\mathcal{P}}_n\right)$ leads to

${\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla u_n\right)\nabla \left(T_k\left(u_n-\phi \right)\right)\text{d}x={\displaystyle {\int }_{\Omega }}{f}_n{T}_k\left(u_n-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u_n-\phi \right)\text{d}x\mathrm{,}$ (3.17)

i.e.,

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla u_n\right)\nabla T_k\left(u_n-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla \phi \right)\nabla T_k\left(u_n-\phi \right)\text{d}x\\ -{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla \phi \right)\nabla T_k\left(u_n-\phi \right)\text{d}x={\displaystyle {\int }_{\Omega }}{f}_n{T}_k\left(u_n-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u_n-\phi \right)\text{d}x\mathrm{,}\end{array}$ (3.18)

which implies that

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}\left(a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla u_n\right)-a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla \phi \right)\right)\nabla T_k\left(u_n-\phi \right)\text{d}x\\ +{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla \phi \right)\nabla T_k\left(u_n-\phi \right)\text{d}x={\displaystyle {\int }_{\Omega }}{f}_n{T}_k\left(u_n-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u_n-\phi \right)\text{d}x\mathrm{.}\end{array}$ (3.19)

Thanks to assumption (1.5) and the definition of truncation function, we have

${\displaystyle {\int }_{\Omega }}\left(a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla u_n\right)-a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla \phi \right)\right)\nabla T_k\left(u_n-\phi \right)\text{d}x\ge 0.$ (3.20)

Combining (3.19) and (3.20), we obtain (3.16).

Step 3: We claim that

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}u\mathrm{,}\nabla \phi \right)\nabla T_k\left(u-\phi \right)\text{d}x\\ \le {\displaystyle {\int }_{\Omega }}f{T}_k\left(u-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u-\phi \right)\text{d}x\forall \phi \in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\cap L^{\infty }\left(\Omega \right)\mathrm{.}\end{array}$ (3.21)

Let $M=k+{\Vert \phi \Vert }_{\infty }$. Since $T_M\left(u_n\right)$ converges to $T_M\left(u\right)$ weakly in $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, then

$T_k\left(u_n-\phi \right)\rightharpoonup T_k\left(u-\phi \right)\text{weakly}\text{in}{W}_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\mathrm{.}$ (3.22)

Thanks to assumption (1.4), we have

$\begin{array}{l}{\left|a\left(x\mathrm{,}{T}_M\left(u_n\right)\mathrm{,}\nabla \phi \right)\right|}^{p^{\prime }\left(x\right)}{\nu }^{\frac{p^{\prime }\left(x\right)}{p\left(x\right)}}\\ \le \beta {\left[b\left(x\right)+{\left|T_M\left(u_n\right)\right|}^{p\left(x\right)-1}+{\nu }^{\frac{1}{p^{\prime }\left(x\right)}}{\left(\gamma \left(T_M\left(u_n\right)\right)\left|\nabla \phi \right|\right)}^{p\left(x\right)-1}\right]}^{p^{\prime }\left(x\right)}\\ \le C\left[b{\left(x\right)}^{p^{\prime }\left(x\right)}+{\left|T_M\left(u_n\right)\right|}^{p\left(x\right)}+\nu \left(x\right){\gamma }_0^{p\left(x\right)}{\left|\nabla \phi \right|}^{p\left(x\right)}\right]\mathrm{,}\end{array}$ (3.23)

where ${\gamma }_0=\sup \left\{\left|\gamma \left(s\right)\right|\mathrm{<mo>\; :\; </mo>}\left|s\right|\le k+{\Vert \phi \Vert }_{\infty }\right\}$ and C is a positive constant. Since $T_M\left(u_n\right)$ converges to $T_M\left(u\right)$ weakly in $W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$, strongly in $L^{p\left(x\right)}\left(\Omega \right)$ and a.e. in $\Omega$, thus

${\left|a\left(x\mathrm{,}{T}_M\left(u_n\right)\mathrm{,}\nabla \phi \right)\right|}^{p^{\prime }\left(x\right)}{\nu }^{\frac{p^{\prime }\left(x\right)}{p\left(x\right)}}\to {\left|a\left(x\mathrm{,}{T}_M\left(u\right)\mathrm{,}\nabla \phi \right)\right|}^{p^{\prime }\left(x\right)}{\nu }^{\frac{p^{\prime }\left(x\right)}{p\left(x\right)}}\text{a}\text{.e}\text{in}\Omega$

and

$\begin{array}{l}C\left[b{\left(x\right)}^{p^{\prime }\left(x\right)}+{\left|T_M\left(u_n\right)\right|}^{p\left(x\right)}+\nu \left(x\right){\gamma }_0^{p\left(x\right)}{\left|\nabla \phi \right|}^{p\left(x\right)}\right]\\ \to C\left[b{\left(x\right)}^{p^{\prime }\left(x\right)}+{\left|T_M\left(u\right)\right|}^{p\left(x\right)}+\nu \left(x\right){\gamma }_0^{p\left(x\right)}{\left|\nabla \phi \right|}^{p\left(x\right)}\right]\mathrm{.}\end{array}$

Combining (3.21), (3.22) and using Vitali’s theorem, we obtain

${\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}{u}_n\mathrm{,}\nabla \phi \right)\nabla T_k\left(u_n-\phi \right)\text{d}x\to {\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}u\mathrm{,}\nabla \phi \right)\nabla T_k\left(u-\phi \right)\text{d}x\mathrm{.}$ (3.24)

Now, we show that

${\displaystyle {\int }_{\Omega }}{f}_n{T}_k\left(u_n-\phi \right)\text{d}x\to {\displaystyle {\int }_{\Omega }}f{T}_k\left(u-\phi \right)\text{d}x\mathrm{.}$ (3.25)

In the first time, we have $f_n{T}_k\left(u_n-\phi \right)\to f{T}_k\left(u-\phi \right)$ a.e in $\Omega$, $\left|f_n{T}_k\left(u_n-\phi \right)\right|\le k\left|f_n\right|$ and $k\left|f_n\right|\to k\left|f\right|$ in $L^1\left(\Omega \right)$. In the second time, by using Vitali’s theorem we obtain (3.25).

Since $F\in {\left(L^{p^{\prime }\left(x\right)}\left(\Omega \mathrm{,}{\nu }^{\ast }\right)\right)}^N$, one has

${\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u_n-\phi \right)\text{d}x\to {\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u-\phi \right)\text{d}x\mathrm{.}$ (3.26)

Thanks to (3.24), (3.25) and (3.26), we obtain (3.21).

Step 4: In this step, we introduce the following generalization of Minty’s lemma in weighted Sobolev space with variable exponents $W^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ (which is proved in [15] ).

Lemma 3.1. ( [15] ) Let u be a measurable function such that $T_k\left(u\right)\in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)$ for every $k>0$ . Then the following statements are equivalent:

1) ${\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}u\mathrm{,}\nabla \phi \right)\nabla T_k\left(u-\phi \right)\text{d}x\le {\displaystyle {\int }_{\Omega }}f{T}_k\left(u-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u-\phi \right)\text{d}x\mathrm{,}$

2) ${\displaystyle {\int }_{\Omega }}a\left(x\mathrm{,}u\mathrm{,}\nabla u\right)\nabla T_k\left(u-\phi \right)\text{d}x={\displaystyle {\int }_{\Omega }}f{T}_k\left(u-\phi \right)\text{d}x+{\displaystyle {\int }_{\Omega }}F\nabla T_k\left(u-\phi \right)\text{d}x\mathrm{,}$

for every $\phi \in W_0^{\mathrm{1,}p\left(x\right)}\left(\Omega \mathrm{,}\nu \right)\cap L^{\infty }\left(\Omega \right)$ and for every $k>0$ .

Finally, the result (3.21) and the lemma 3.1 lead to the completion of the proof of theorem 3.1.

4. Conclusion

In this article, we have demonstrated the existence of a solution of a problem with a second measure member and in the space of Sobolev with variable exponent using Minty’s lemma. It is a very important technique in which we use the notions of hemicontinuous and pseudo-monotonic instead of broad or strict monotony.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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