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   PDF    HTML   XML   416 Downloads   1,537 Views   Citations

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

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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.