Abdelkrim Barbara, El Houcine Rami, Elhoussine Azroul
Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, Atlas Fez, Morocco.
DOI: 10.4236/am.2021.126035   PDF    HTML   XML   420 Downloads   1,567 Views   Citations

Abstract

We consider, for a bounded open domain Ω in Rⁿ and a function u : Ω → R^m, the quasilinear elliptic system: (1). We generalize the system (QES)_(f,g) in considering a right hand side depending on the jacobian matrix Du. Here, the star in (QES)_(f,g) indicates that f may depend on Du. In the right hand side, v belongs to the dual space W^-1,P’(Ω, ω^*, R^m), , f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ, but with only very mild monotonicity assumptions.

Keywords

Quasilinear Elliptic, Sobolev Spaces with Weight, Young Measure, Galerkin Scheme

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1. Introduction

In this paper, the main point is that we do not require monotonicity in the strict monotonicity of a typical Leray-Lions operator as it is usually assumed in previous papers. The aims of this text are to prove analogous existence results under relaxed monotonicity, in particular under strict quasi-monotonicity. The main technical tool we advocate and use throughout the proof is Young measures. By applying a Galerkin schema, we obtain easily an approximating sequence $u_k$. The Ball theorem [1] and especially the resulting tool mode available by Hungerbühler to partial differential equation theory give them a sufficient control on the gradient approximating sequence $D{u}_k$ to pass to the limit. This method is used by Dolzmann [2], G. J. Minty [3], H. Brezis [4], H. E. Stromberg [5], Muller [6], J. L. Lions [7], Kristznsen, J. Lower [8], M. I. Visik [9] and mainly by Hungurbühler to get the existence of a weak solution for the quasi-linear elliptic system [10]. This paper can be seen as generalization of Hungurbühler and as a continuation of Y-Akdim [11].

This kind of problems finds its applications in the model of Thomas-Fermis in atomic physics [12], and also porous flow modeling in reservoir [13].

2. Priliminaries

Let $\omega =\left\{{\omega }_{ij}\mathrm{<mo>\; ;\; </mo>0}\le i\le n\mathrm{<mo>\; ;\; </mo>1}\le j\le m\right\}$ the weight function systems defined in $\Omega$ and satisfying the following integrability conditions:

$\begin{array}{l}{\omega }_{ij}\in L_{loc}^1\left(\Omega \right)\mathrm{,}{\omega }_{ij}^{\frac{-1}{p-1}}\in L_{loc}^1\left(\Omega \right)\mathrm{,}\text{for}\text{some}p\in \left]\mathrm{1,}\infty \right[\\ \text{and}\exists s>0\text{such}\text{that}{\omega }_{ij}^{-s}\in L^1\left(\Omega \right)\mathrm{.}\end{array}$ (2.1)

with ${\omega }^{*}=\left\{{\omega }_{ij}^{*}={\omega }_{ij}^{1-p^{\prime }},0\le i\le n,1\le j\le m\right\}$, $\sigma =\left({\sigma }_{rs}\right)$ with $1\le s\le n\mathrm{,}1\le r\le m$ and which satisfies some hypotheses (see below).

We denote by $I{M}^{m\times n}$ the real vector space of $m\times n$ matrices equipped with

the inner product $M:N={\displaystyle \underset{ij}{\sum }{M}_{ij}{N}_{ij}}$.

The Jacobian matrix of a function $u\mathrm{<mo>\; :\; </mo>}\Omega \to I{R}^m$ is denoted by $Du\left(x\right)=\left(D_{1}u\left(x\right)\mathrm{,}{D}_{2}u\left(x\right)\mathrm{,}\cdots \mathrm{,}{D}_{n}u\left(x\right)\right)$ with $D_i=\partial /\partial \left(x_i\right)$.

The space $W^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ is the set of functions

$\begin{array}{l}\left\{u=u\left(x\right)/u\in L^p\left(\Omega ,\overline{{\omega }_0},I{R}^m\right)\right\},D_{ij}u=\frac{\partial u^i}{\partial x_j}\in L^p\left(\Omega ,{\omega }_{ij},I{R}^m\right),\\ 1\le i\le n,1\le j\le m.\end{array}$

with

$L^p\left(\Omega \mathrm{,}{\omega }_{ij}\mathrm{,}I{R}^m\right)=\left\{u=u\left(x\right)\mathrm{<mo>\; /\; </mo>}\left|u\right|{\omega }_{ij}^{\frac{1}{p}}\in L^p\left(\Omega \mathrm{,}I{R}^m\right)\right\}$

The weighted space $W^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ can be equipped by the norm:

${\Vert u\Vert }_{1,p,\omega }={\left({\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|u_j\right|}^p{\omega }_{0j}\text{d}x+{\displaystyle \underset{1\le i\le n,1\le j\le m}{\sum }}{\displaystyle {\int }_{\Omega }}{\left|D_{ij}u\right|}^p{\omega }_{ij}\text{d}x\right)}^{\frac{1}{p}}$

where $\overline{{\omega }_0}=\left({\omega }_{0j}\right)$ and $1\le j\le m$. the norm ${\Vert \mathrm{.}\Vert }_{\mathrm{1,}\omega \mathrm{,}p}$ is equivalent to the norm $\left|\right||\cdot |\left|\right|$, on $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$, such that, $\left|\right|\left|u\right|\left|\right|={\left({\displaystyle {\sum }_{1\le i\le n,1\le j\le m}}{\displaystyle {\int }_{\Omega }}{\left|D_{ij}u\right|}^p{\omega }_{ij}\text{d}x\right)}^{\frac{1}{p}}$.

Proposition 2.1 The weighted Sobolev space $W^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ is a Banach space, separable and reflexive. The weighted Sobolev space $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ is the closure of $C_0^{\infty }\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ in $W^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ equipped by the norm ${\Vert \mathrm{.}\Vert }_{\mathrm{1,}p\mathrm{,}\omega }$ .

Proof: The prove of proposition is a slight modification of the analogous one in [14] [Kufner-Drabek].

Definition 2.1 A Young measure ${\left({\vartheta }_x\right)}_{x\in \Omega }$ is called $W^{\mathrm{1,}p}$ -gradient young measures ( $1\le p<\infty$ ) if it is associated to a sequence of gradients $D{u}_k$ such that $u_k$ is bounded in $W^{\mathrm{1,}p}\left(\Omega \right)$ . The $W^{\mathrm{1,}p}$ -gradient young measures ${\left({\vartheta }_x\right)}_{x\in \Omega }$ is called homogeneous, if it doesn’t depend on x, i-e, if ${\vartheta }_x=\vartheta$ for a.e. $x\in \Omega$.

Theorem 2.1 (Kinder Lehirer-Pedregal) let ${\left({\upsilon }_x\right)}_{x\in \Omega }$ , be a family of probability measures in ${\left(C\left(M^{m\times n}\right)\right)}^{\prime }$ . Then, ${\left({\upsilon }_x\right)}_{x\in \Omega }$ is $W^{\mathrm{1,}p}$ Young measures if and only if:

1) There is a $u\in W^{\mathrm{1,}p}\left(\Omega \mathrm{,}I{R}^m\right)$ such that $Du\left(x\right)={\displaystyle {\int }_{M^{m\times n}}}A\text{d}{\vartheta }_x\left(A\right)$, a.e in $\Omega$.

2) Jensen’s inequality: $\varphi \left(Du\left(x\right)\right)\le {\displaystyle {\int }_{M^{m\times n}}}\varphi \left(A\right)\text{d}{\vartheta }_x\left(A\right)$ hold for all $\varphi \in X^p$ quasi-convex, and.

3) The function: $\psi \left(x\right)={\displaystyle {\int }_{M^{m\times n}}}{\left|A\right|}^p\text{d}{\vartheta }_x\left(A\right)\in L^1\left(\Omega \right)$. Here, $X^p$ denotes the (not separable) space: $X^p=\left\{\psi \in C\left(M^{m\times n}\right)\mathrm{<mo>\; :\; </mo>}\left|\psi \left(A\right)\right|\le c\times \left(1+{\left|A\right|}^p\right),\text{forall}A\in M^{m\times n}\right\}$.

proof: see [15].

Theorem 2.2 (Ball) Let $\Omega \in I{R}^n$ be Lebesgue measurable, let $K\in I{R}^m$ be closed, and let $u_j\mathrm{<mo>\; :\; </mo>}\Omega \to I{R}^m\mathrm{,}j\in IN$ , be a sequence of Lebesgue measurable functions satisfying $u_j\to K$ , as $j\to \infty$, i.e. given any open neighborhood U of $K\in I{R}^m$ , ${\lim }_{j\to \infty }\left|x\in \Omega \mathrm{<mo>\; :\; </mo>}{u}_j\left(x\right)\in U\right|=0$. Then there existsa subsequence $u_k$ of $u_j$ and a family ${\vartheta }_x\mathrm{,}x\in \Omega$, of positive measures on $I{R}^m$ , depending measurablyon x, such that

1) ${\Vert {\vartheta }_x\Vert }_M={\displaystyle {\int }_{I{R}^m}}\text{d}{\vartheta }_x\le 1$, for a.e $x\in \Omega$.

2) $supp{\vartheta }_x\subset K$ for a.e $x\in \Omega$.

3) $f\left(u_k\right){\rightharpoonup }^{\ast }\langle {\vartheta }_x\mathrm{,}f\rangle ={\displaystyle {\int }_{I{R}^m}}f\left(\lambda \right)\text{d}{\vartheta }_x\left(\lambda \right)$ in $L^{\infty }\left(\Omega \right)$ for each continuous functions $f\mathrm{<mo>\; :\; </mo>}I{R}^m\to IR$ satisfying

$\lim f\left(\lambda \right)=\mathrm{0,}\left|\lambda \right|\to \infty$ [1].

Theorem 2.3 (vitali) Let $\Omega \in I{R}^n$ be an open bounded domain and let $u_n$ be a sequence in $L^p\left(\Omega \mathrm{,}I{R}^m\right)$ with $1\le p<\infty$ .

Then $u_n$ is a cauchy sequence in the $L^p$ -norm if and only if the two following conditions hold:

1) $u_n$ is cauchy in measure (i.e. $\forall \epsilon >0$, $\left|\left\{x\in \Omega ,\left|u_n\left(x\right)-u_m\left(x\right)\right|\ge \epsilon \right\}\right|=0$ as $m\mathrm{,}n\to \infty$ .

2) ${\left|u_n\right|}^p$ is equiintegrable i.e.:

( $su{p}_n{\displaystyle {\int }_{\Omega }}{\left|u_n\right|}^p\text{d}x<\infty$ and $\forall \epsilon >0,\exists \delta >0$ such that ${\displaystyle {\int }_E}{\left|u_n\right|}^p\text{d}x<\epsilon$ for all n whenever $E\subset \Omega$ and $\left|E\right|<\delta$ ). Note that if $u_n$ converges pointiest, then $u_n$ is cauchy in measure.

Hypotheses (H₀) (Hardy-Type inequalities): There exist some constant $c>0$, some weighted function $\gamma$ and some real q ( $1<q<\infty$ ) such that,

${\left({\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|u_j\left(x\right)\right|}^q{\gamma }_j\left(x\right)\text{d}x\right)}^{\frac{1}{q}}\le c{\left({\displaystyle \underset{1\le i\le n,1\le j\le m}{\sum }}{\displaystyle {\int }_{\Omega }}{\left|D_{ij}u\right|}^p{\omega }_{ij}\right)}^{\frac{1}{p}},$

for all $u\in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$, with $\gamma =\left\{{\gamma }_j/1\le j\le m\right\}$.

The injection $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ ↪↪ $L^q\left(\Omega \mathrm{,}\gamma \mathrm{,}I{R}^m\right)$ is compact, and $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ ↪↪ $L^r\left(\Omega \mathrm{,}I{R}^m\right)$ is compact, (by [14] ) with

$\{\begin{array}{ll}1\le r\prec \frac{nps}{n\left(s+1\right)-ps}\hfill & \text{if}ps\prec n\left(s+1\right)\hfill \\ r\ge 1\hfill & \text{if}n\left(s+1\right)\prec ps\hfill \end{array}$

(H₁) Continuity: $\sigma \mathrm{<mo>\; :\; </mo>}\Omega \times I{R}^m\times I{M}^{m\times n}\to I{M}^{m\times n}$ is a Carathéodory function (i-e $x\mapsto \sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is measurable for every $\left(u\mathrm{,}F\right)\in I{R}^m\times I{M}^{m\times n}$ and $\left(u\mathrm{,}F\right)\mapsto \sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is continuous for almost every $x\in \Omega$ ). (H₂) Growths and coercivity conditions: There exist $c_1\ge 0$, $c_2>0$, ${\lambda }_1\in L^{p^{\prime }}\left(\Omega \right)$, ${\lambda }_2\in L^1\left(\Omega \right)$, ${\lambda }_3\in L^{(p/\alpha {)}^{\prime }}\left(\Omega \right)$, $0<\alpha <p$, $1<q<\infty$ and $\beta >0$ such that for all $1\le r\le n$, $1\le s\le m$, we have:

$\left|{\sigma }_{rs}\left(x,u,F\right)\right|\le \beta w_{rs}^{1/p}\left[{\lambda }_1\left(x\right)+c_1{\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\left|{\gamma }_j\right|}^{1/p^{\prime }}\cdot {\left|u_j\right|}^{q/p^{\prime }}+c_1{\displaystyle \underset{1\le i\le N,1\le j\le m}{\sum }}{\omega }_{ij}^{1/p^{\prime }}{\left|F_{ij}\right|}^{p-1}\right]$ (2.2)

and

$\sigma \left(x,u,F\right):F\ge -{\lambda }_2\left(x\right)-{\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\omega }_{0j}{\left(x\right)}^{\alpha /p}{\lambda }_3\left(x\right){\left|u_j\right|}^{\alpha }+c_2{\displaystyle \underset{1\le i\le n,1\le j\le m}{\sum }}{\omega }_{ij}\left(x\right)\cdot {\left|F_{ij}\right|}^p$ (2.3)

(H₃) Monotonicity conditions: $\sigma$ satisfies one of the following conditions:

1) For all $x\in \Omega$, and all $u\in I{R}^m$, the map $F\mapsto \sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is a $C^1$ -function and is monotone (i-e, $\left(\sigma \left(x\mathrm{,}u\mathrm{,}F\right)-\sigma \left(x\mathrm{,}u\mathrm{,}G\right)\right)\mathrm{<mo>\; :\; </mo>}\left(F-G\right)\ge 0$, for all $x\in \Omega$, all $u\in I{R}^m$ and all $F\mathrm{,}G\in I{M}^{m\times n}$ ).

2) There exists a function $W\mathrm{<mo>\; :\; </mo>}\Omega \times I{R}^m\times I{M}^{m\times n}\to I{M}^{m\times n}$ such that $\sigma \left(x\mathrm{,}u\mathrm{,}F\right)=\frac{\partial W}{\partial F}\left(x\mathrm{,}u\mathrm{,}F\right)$ and $F\mapsto W\left(x\mathrm{,}u\mathrm{,}F\right)$ is convex and $C^1$ function.

3) For all $x\in \Omega$, and for all $u\in I{R}^m$ the map $F\mapsto \sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is strictly monotone (i.e., $\sigma \left(x\mathrm{,}u\mathrm{,.}\right)$ is monotone and: $\left[\left(\sigma \left(x,u,F\right)-\sigma \left(x,u,G\right)\right):\left(F-G\right)=0\right]\Rightarrow F=G$ ).

4) $\sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is strictly p-quasi-monotone in F, i.e.,

${\displaystyle {\int }_{I{M}^{m\times n}}}\left(\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)-\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -\bar{\lambda }\right)\text{d}\vartheta \left(\lambda \right)>\mathrm{0,}$

for all homogeneous $W^{\mathrm{1,}p\mathrm{,}w}$ -gradient young measures $\vartheta$ with center of mass $\bar{\lambda }=\langle \vartheta ,id\rangle$ which are not a single Dirac mass.

The main point is that we do not require strict monotonicity or monotonicity in the variables $\left(u\mathrm{,}F\right)$ in (H₃) as it is usually assumed in previous work (see [15] or [16] ).

${\left(F_0\right)}^{\ast }$ : (continuity) $f\mathrm{<mo>\; :\; </mo>}\Omega \times I{R}^m\times I{M}^{m\times n}\to I{R}^m$ is a Carathéodory function i-e: $x\mapsto f\left(x\mathrm{,}u\mathrm{,}F\right)$ is measurable for every $u\in I{R}^m$, and $F\in I{M}^{m\times n}$, $\left(u\mathrm{,}F\right)\mapsto f\left(x\mathrm{,}u\mathrm{,}F\right)$ is continuous for almost every $x\in \Omega$.

${\left(F_1\right)}^{\ast }$ : (growth condition): The exist: $b_1\in L^{p^{\prime }}\left(\Omega \right)$, ${c^{\prime }}_1>0$, ${c^{\prime }}_2>0$ such that:

$\left|f_j\left(x\mathrm{,}u\right)\right|\le \left[b_1\left(x\right)+{c^{\prime }}_1{\gamma }_j^{\frac{1}{p^{\prime }}}{\left|u_j\right|}^{\frac{q}{p^{\prime }}}+{c^{\prime }}_2{\displaystyle \underset{r\mathrm{,}s}{\sum }}{\omega }_{rs}^{\frac{1}{P^{\prime }}}{\left|F_{rs}\right|}^{p-1}\right]{\omega }_{0j}^{\frac{1}{P}}\mathrm{<mo>\; ;\; </mo>}$

$\forall 1\le j\le m$ ${\left(G_0\right)}^{\ast }$ : (continuity) the map $g\mathrm{<mo>\; :\; </mo>}\Omega \times I{R}^m\to I{M}^{m\times n}$ is a Carathéodory function. ${\left(G_1\right)}^{\ast }$ : (growth condition) There exist: $b_2\in L^{p^{\prime }}(\; \Omega \; )$

$\left|g_{rs}\right|\le {\omega }_{rs}^{\frac{1}{p}}\left[b_2+{\displaystyle \underset{j}{\sum }}{\gamma }_j^{\frac{1}{p^{\prime }}}{\left|u_j\right|}^{\frac{q}{p^{\prime }}}\right]$

For all $1\le r\le n$ and $1\le s\le m$.

Our aim of this paper is to prove the existence of the problem ${\left(QES\right)}_{f\mathrm{,}g}$ in the space $W_0^{\mathrm{1,}P}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$.

Remark 2.1 -The condition ${\left(F_0\right)}^{\ast }$ and $\left(G_0\right)$ ensure the measurability of f and g for all measurable function u.

- $\left(F_1\right)$ and ${\left(G_1\right)}^{\ast }$ ensure that growths conditions, in particularly: if $u\in W_0^{\mathrm{1,}P}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ then $f\left(\mathrm{.,}u\left(\mathrm{.}\right)\mathrm{,}D\left(\mathrm{.}\right)\right)\cdot u\left(\mathrm{.}\right)$ and $g\left(\mathrm{.,}u\right)\mathrm{<mo>\; :\; </mo>}Du$ is in $L^1\left(\Omega \mathrm{,}\omega \right)$.

- Exploiting the convergence in measure of the gradients of the approximating solutions, we will prove the following theorem.

Theorem 2.4 If $p\in \left(\mathrm{1,}\infty \right)$ and $\sigma$ satisfies the conditions (H₀)-(H₃), then the Dirichlet problem ${\left(QES\right)}_{f\mathrm{,}g}^{\ast }$ has a weak solution $u\in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ , for every $v\in W^{-\mathrm{1,}{p}^{\prime }}\left(\Omega \mathrm{,}{\omega }^{\ast }\mathrm{,}I{R}^m\right)$ , f satisfies ${\left(F_0\right)}^{\ast }$ and ${\left(F_1\right)}^{\ast }$ and g satisfies $\left(G_0\right)$ and $\left(G_1\right)$ .

In order to prove theorems, we will apply a Galerkin scheme, with this aim in view, we establish in the following subsections, the key ingredient to pass to the limit for this, we assume that the conditions: (H₀)-(H₃), ${\left(F_0\right)}^{\ast }$, ${\left(F_1\right)}^{\ast }$, $\left(G_0\right)$ and $\left(G_1\right)$.

Lemma 2.1 For arbitrary $u\in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ and $v\in W^{-\mathrm{1,}{p}^{\prime }}\left(\Omega \mathrm{,}{\omega }^{\ast }\mathrm{,}I{R}^m\right)$ , the functional

$\begin{array}{l}F\left(u\right)\mathrm{<mo>\; :\; </mo>}{W}_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)\to IR\\ \phi \mapsto {\displaystyle {\int }_{\Omega }}\sigma \left(x\mathrm{,}u\left(x\right)\mathrm{,}Du\left(x\right)\right)\mathrm{<mo>\; :\; </mo>}D\phi \left(x\right)\text{d}x-\langle v\mathrm{,}\phi \rangle \\ -{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}\phi \text{d}x+{\displaystyle {\int }_{\Omega }}g\left(x\mathrm{,}u\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{d}x\mathrm{.}\end{array}$

is well defined, linear and bounded.

Proof For all $\phi \in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$, we denote

$F\left(u\right)\left(\phi \right)=I_1+I_2+I_3+I_4$

with

$I_1={\displaystyle {\int }_{\Omega }}\sigma \left(x,u\left(x\right),Du\left(x\right)\right):D\phi \left(x\right)\text{d}x,$

and

$I_2=-\langle v,\phi \rangle .$

$I_3=-{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}\phi \text{d}x$

$I_4={\displaystyle {\int }_{\Omega }}g\left(x\mathrm{,}u\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{d}x$

We define

$I_{rs}={\displaystyle {\int }_{\Omega }}{\sigma }_{rs}\left(x,u\left(x\right),Du\left(x\right)\right):D_{rs}\phi \left(x\right)\text{d}x$

Firstly, by virtue of the growth conditions (H₂) and the Hölder inequality, one has

$\begin{array}{c}\left|I_{rs}\right|\le {\displaystyle {\int }_{\Omega }}\left|{\sigma }_{rs}\left(x,u\left(x\right),Du\left(x\right)\right)\right|:\left|D_{rs}\phi \left(x\right)\right|\text{d}x\\ \le {\displaystyle {\int }_{\Omega }}\beta {\omega }_{rs}^{1/p}\left(x\right)[{\lambda }_1\left(x\right)+c_1{\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\left|{\gamma }_j\left(x\right)\right|}^{1/p^{\prime }}{\left|u_j\left(x\right)\right|}^{q/p^{\prime }}\\ +c_1{\displaystyle \underset{1\le i\le n,1\le j\le m}{\sum }}{\omega }_{ij}^{1/p^{\prime }}{\left|D_{ij}u\right|}^{p-1}]\left|D_{rs}\phi \right|\text{d}x\\ \le \beta [{\left({\displaystyle {\int }_{\Omega }}{\left|{\lambda }_1\left(x\right)\right|}^{p^{\prime }}\text{d}x\right)}^{1/p^{\prime }}{\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\phi \left(x\right)\right|}^p{\omega }_{rs}\text{d}x\right)}^{1/p}\\ +{\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\phi \left(x\right)\right|}^p{\omega }_{rs}\right)}^{1/p}{\left({\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|u_j\right|}^q{\gamma }_j\text{d}x\right)}^{1/p^{\prime }}\\ +{\left({\displaystyle \underset{1\le i\le n,1\le j\le m}{\sum }}{\displaystyle {\int }_{\Omega }}{\left|D_{ij}u\right|}^p{\omega }_{ij}\text{d}x\right)}^{1/p^{\prime }}{\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\phi \right|}^p{\omega }_{rs}\text{d}x\right)}^{1/p}]\end{array}$

with $\left(p=p^{\prime }\left(p-1\right)\right)$, and thanks to Hardy inequality we have:

$\begin{array}{c}\left|I_{rs}\right|\le c\beta \left[{\Vert {\lambda }_1\Vert }_{p^{\prime }}{\Vert \phi \Vert }_{\mathrm{1,}p\mathrm{,}{\omega }_{rs}}+c_1{\Vert D\phi \Vert }_{p\mathrm{,}{\omega }_{rs}}{\left({\displaystyle {\int }_{\Omega }}{\left|u\right|}^q\gamma \text{d}x\right)}^{\mathrm{1\; <mo>\; /\; </mo>}{p}^{\prime }}+c_1{\displaystyle \underset{ij}{\sum }}{\Vert D\phi \Vert }_{p\mathrm{,}{\omega }_{ij}}{\Vert Du\Vert }_{p\mathrm{,}{\omega }_{rs}}\right]\\ \le c^{\prime }\beta \left[{\Vert {\lambda }_1\Vert }_{p^{\prime }}{\Vert \phi \Vert }_{\mathrm{1,}p\mathrm{,}{\omega }_{rs}}+{\Vert \phi \Vert }_{\mathrm{1,}p\mathrm{,}{\omega }_{rs}}{\Vert u\Vert }_{q\mathrm{,}\gamma }+{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }{\Vert \phi \Vert }_{\mathrm{1,}p\mathrm{,}{\omega }_{rs}}\right]\end{array}$

with $c^{\prime }=\max \left(c\mathrm{,1}\right)$. Which gives

$\left|I_1\right|\le c^{\prime }\beta \left[{\Vert {\lambda }_1\Vert }_{p^{\prime }}+{\Vert u\Vert }_{1,p,\omega }^{q/p^{\prime }}+{\Vert u\Vert }_{1,p,\omega }\right]{\Vert \phi \Vert }_{1,p,\omega }<\infty .$

and

$\left|I_2\right|\le {\displaystyle {\int }_{\Omega }}\left|v\right|\left|\phi \right|\text{d}x\le {\Vert v\Vert }_{-1,p^{\prime },{\omega }^{*}}{\Vert \phi \Vert }_{1,p,\omega }<\infty .$

$I_3={\displaystyle \underset{j}{\sum }}{\displaystyle {\int }_{\Omega }}{f}_j\left(x,u,Du\right){\phi }_j\left(x\right)\text{d}x$

We denote $I_{\mathrm{3,}j}=\left|{\displaystyle {\int }_{\Omega }}{f}_j\left(x\mathrm{,}u\mathrm{,}Du\right){\phi }_j\left(x\right)\text{d}x\right|$.

$\begin{array}{c}{I}_{\mathrm{3,}j}\le {\displaystyle {\int }_{\Omega }}\left|f_j\left(x\mathrm{,}u\mathrm{,}Du\right)\right|\left|{\phi }_j\left(x\right)\right|\text{d}x\\ \le {\displaystyle {\int }_{\Omega }}{b}_1\left(x\right)\left|{\phi }_j\left(x\right)\right|{\omega }_{0j}^{\frac{1}{P}}\text{d}x+{c^{\prime }}_1{\displaystyle {\int }_{\Omega }}{\gamma }_j^{\frac{1}{p^{\prime }}}{\left|u_j\right|}^{\frac{q}{p^{\prime }}}\left|{\phi }_j\left(x\right)\right|{\omega }_{0j}^{\frac{1}{P}}\\ +{c^{\prime }}_2{\displaystyle {\int }_{\Omega }}{\displaystyle \underset{rs}{\sum }}\left|{\phi }_j\left(x\right)\right|{\left|D_{rs}u\right|}^{p-1}{\omega }_{0j}^{\frac{1}{P}}\text{d}x\\ \le {\left({\displaystyle {\int }_{\Omega }}{\left|b_1\left(x\right)\right|}^{p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}{\left({\displaystyle {\int }_{\Omega }}{\left|{\phi }_j\left(x\right)\right|}^p{\omega }_{0j}\text{d}x\right)}^{\frac{1}{p}}\\ +{\left({\displaystyle {\int }_{\Omega }}{\gamma }_j\left(x\right){\left|u_j\right|}^q\text{d}x\right)}^{\frac{1}{p^{\prime }}}{\left({\displaystyle {\int }_{\Omega }}{\left|{\phi }_j\left(x\right)\right|}^p{\omega }_{0j}\text{d}x\right)}^{\frac{1}{p}}\\ +{\displaystyle \underset{rs}{\sum }}{\left({\displaystyle {\int }_{\Omega }}{\left|{\phi }_j\right|}^p{\omega }_{0j}\right)}^{\frac{1}{p}}\cdot {\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}u\right|}^{p^{\prime }(p-1)}{\omega }_{rs}\right)}^{\frac{1}{p^{\prime }}}\\ \le {\Vert b_1\Vert }_{p^{\prime }}{\Vert \phi \Vert }_{1,p,\omega }+{c^{\prime }}_1{\left({\displaystyle \underset{j}{\sum }}{\displaystyle {\int }_{\Omega }}{\gamma }_j\left(x\right){\left|u_j\right|}^q\text{d}x\right)}^{\frac{1}{p^{\prime }}}\cdot {\Vert \phi \Vert }_{1,p,\omega }+{c^{\prime }}_2{\Vert \phi \Vert }_{1,\omega ,p}\cdot {\Vert Du\Vert }_{1,p,\omega }^{\frac{p}{p^{\prime }}}\end{array}$

$\begin{array}{c}\le {\Vert b_1\Vert }_{p^{\prime }}{\Vert \phi \Vert }_{1,p,\omega }+{c^{\prime }}_1\cdot {\Vert Du\Vert }_{1,p,\omega }\cdot {\Vert \phi \Vert }_{1,p,\omega }+{c^{\prime }}_2{\Vert \phi \Vert }_{1,\omega ,p}\cdot {\Vert Du\Vert }_{1,p,\omega }^{\frac{p}{p^{\prime }}}\\ \le \left(\Vert b_1\Vert +{c^{\prime }}_1\cdot {\Vert Du\Vert }_{1,p,\omega }+{c^{\prime }}_2\cdot {\Vert Du\Vert }_{1,p,\omega }^{\frac{p}{p^{\prime }}}\right)\cdot {\Vert \phi \Vert }_{1,p,\omega }.\end{array}$

$I_4={\displaystyle \underset{rs}{\sum }}{\displaystyle {\int }_{\Omega }}{g}_{rs}\left(x,u\right)D_{rs}\phi \text{d}x$

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}\left|g_{rs}\right|:\left|D_{rs}\phi \right|\text{d}x\\ \le {\displaystyle {\int }_{\Omega }}{b}_2{\omega }_{rs}^{\frac{1}{p}}\cdot D_{rs}\phi \text{d}x+{\displaystyle \underset{j}{\sum }}{\displaystyle {\int }_{\Omega }}{\gamma }_j^{\frac{1}{p^{\prime }}}\left(x\right){\left|u_j\right|}^{\frac{q}{p^{\prime }}}{\omega }_{rs}^{\frac{1}{p}}{D}_{rs}\phi \text{d}x\\ \le {\left({\displaystyle {\int }_{\Omega }}{\left|b_2\right|}^{p^{\prime }}\text{d}x\right)}^{\frac{1}{p^{\prime }}}\cdot {\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\phi \right|}^p{\omega }_{rs}\text{d}x\right)}^{\frac{1}{p}}\\ +{\displaystyle \underset{j}{\sum }}{\left({\displaystyle {\int }_{\Omega }}{\left|u_j\right|}^q{\gamma }_j\left(x\right)\text{d}x\right)}^{\frac{1}{p^{\prime }}}{\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\phi \right|}^p{\omega }_{rs}\left(x\right)\text{d}x\right)}^{\frac{1}{p}}\\ \le {\Vert b_2\Vert }_{p^{\prime }}{\Vert D_{rs}\phi \Vert }_{1,p,{\omega }_{rs}}+{\Vert u\Vert }_{q,\gamma }^{\frac{q}{p^{\prime }}}{\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\phi \right|}^p{\omega }_{rs}\text{d}x\right)}^{\frac{1}{p}}\end{array}$

$\begin{array}{c}{I}_4\le {\Vert b_2\Vert }_{p^{\prime }}{\Vert D_{rs}\phi \Vert }_{1,p,{\omega }_{rs}}+{\Vert u\Vert }_{q,\gamma }^{\frac{q}{p^{\prime }}}{\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\phi \right|}^p{\omega }_{rs}\text{d}x\right)}^{\frac{1}{p}}\\ \le {\Vert b_2\Vert }_{p^{\prime }}\cdot {\Vert D\phi \Vert }_{1,p,\omega }+{\Vert u\Vert }_{q,\gamma }^{\frac{q}{p^{\prime }}}\cdot {\Vert D\phi \Vert }_{1,p,\omega }\\ \le c^{″}{\Vert \phi \Vert }_{1,p,\omega }\end{array}$

Hence $I\le c_4{\Vert \phi \Vert }_{\mathrm{1,}p\mathrm{,}\omega }$. With $c_4<\infty$.

Finally the functional $F\left(\mathrm{.}\right)$ is bounded.

Lemma 2.2 The restriction of F to a finite dimensional linear subspace V of $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ is continuous.

Proof Let d be the dimension of V and $\left(e_1\mathrm{,}{e}_2\mathrm{,}\cdots \mathrm{,}{e}_d\right)$ a basis of V. Let $u_j={\displaystyle \underset{1\le i\le d}{\sum }{a}_j^i\cdot e_i}$ be a sequence in V which converges to $u={\displaystyle \underset{1\le i\le d}{\sum }{a}^i{e}_i}$ in V. The sequence $\left(a_j\right)$ converge to $a\in I{R}^d$, so $u_j\to u$ and $D{u}_j\to Du$ a.e., on the other hand ${\Vert u_j\Vert }_p$ and ${\Vert D{u}_j\Vert }_p$ are bounded by a constant c. Thus, it follows by the continuity conditions (H₁), that

$\sigma \left(x\mathrm{,}{u}_j\mathrm{,}D{u}_j\right)\mathrm{<mo>\; :\; </mo>}D\phi \to \sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}D\phi$

for all $\phi \in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ and a.e. in $\Omega$. Let ${\Omega }^{\prime }$ be a measurable subset of $\Omega$ and let $\phi \in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$.

Thanks to the condition (H₂), we get

${\displaystyle {\int }_{{\Omega }^{\prime }}}\left|\sigma \left(x,u_j,D{u}_j\right):D\phi \right|\text{d}x<\infty ,$

By the continuity conditions ${\left(F_0\right)}^{\ast }$ and $\left(G_0\right)$ we have:

$f\left(x\mathrm{,}{u}_j\mathrm{,}D{u}_j\right)\cdot \phi \to f\left(x\mathrm{,}u\mathrm{,}Du\right)\cdot \phi$

And

$g\left(x\mathrm{,}{u}_j\right)\cdot D\phi \to g\left(x\mathrm{,}u\right)\cdot D\phi$

almost everywhere. Moreover we infer from the growth conditions ${\left(F_1\right)}^{\ast }$ and $\left(G_1\right)$ that the sequences:

$\left(\sigma \left(x\mathrm{,}{u}_j\mathrm{,}D{u}_j\right)\mathrm{<mo>\; :\; </mo>}D\phi \right)$, $\left(f\left(x\mathrm{,}{u}_j\mathrm{,}D{u}_j\right)\cdot \phi \right)$ and $\left(g\left(x\mathrm{,}{u}_j\right)\cdot D\phi \right)$

Are equi-integrable. Indeed, if ${\Omega }^{\prime }\subset \Omega$ is a measurable subset and $\phi \in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ then:

${\displaystyle {\int }_{{\Omega }^{\prime }}}\left|f\left(x,u_j,D{u}_j\right)\cdot \phi \right|\text{d}x<\infty$ (by ${\left(F_1\right)}^{\ast }$ and Hölder inequality).

${\displaystyle {\int }_{{\Omega }^{\prime }}}\left|g\left(x,u_j\right)\cdot D\phi \right|\text{d}x<\infty$ (by $\left(G_1\right)$ and Hölder inequality).

${\displaystyle {\int }_{{\Omega }^{\prime }}}\left|\sigma \left(x\mathrm{,}{u}_j\mathrm{,}D{u}_j\right)\mathrm{<mo>\; :\; </mo>}D\phi \right|\text{d}x<\infty$ (by Hölder inequality).

which implies that $\sigma \left(x\mathrm{,}{u}_j\mathrm{,}D{u}_j\right)\mathrm{<mo>\; :\; </mo>}D\phi$ is equi-integrable. And by applying the Vitali’s theorem, it follows that

${\displaystyle {\int }_{\Omega }}\sigma \left(x\mathrm{,}{u}_j\mathrm{,}D{u}_j\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{d}x\to {\displaystyle {\int }_{\Omega }}\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{d}x\mathrm{,}$

for all $\phi \in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$.

Finally

$\underset{j\to \infty }{\lim }\langle F\left(u_j\right)\mathrm{,}\phi \rangle =\langle F\left(u\right)\mathrm{,}\phi \rangle \mathrm{,}$

which means that

$F\left(u_j\right)\to F\left(u\right)\text{in}{W}^{-\mathrm{1,}{p}^{\prime }}\left(\Omega \mathrm{,}{\omega }^{\ast }\mathrm{,}I{R}^m\right)\mathrm{.}$

Remark 2.2 Now, the problem ${\left(QES\right)}_{f\mathrm{,}g}^{\ast }$ is equivalent to find a solution $u\in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ such that $\langle F\left(u\right),\phi \rangle =0$ , for all $\phi \in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ .

In order tofind such a solution we apply a Galerkin scheme.

3. Galerkin Approximation

Remark 3.1 (Galerkin Schema)

Let $V_1\subset V_2\subset \cdots \subset W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ be a sequence of finite dimensional subspaces with ${\displaystyle {\cup }_{k\in IN}}{V}_k$ dense in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$. The sequence $V_k$ exists since $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ is separable.

Let us fix some k, we assume that $V_k$ has a dimension d and that $\left(e_1\mathrm{,}{e}_2\mathrm{,}\cdots \mathrm{,}{e}_d\right)$ is a basis of $V_k$. Then, we define the map,

$\begin{array}{l}G\mathrm{<mo>\; :\; </mo>}I{R}^k\to I{R}^k\\ \left(a_1\mathrm{,}\cdots \mathrm{,}{a}_k\right)\mapsto \left(\langle F\left(u\right)\mathrm{,}{e}_1\rangle \mathrm{,}\cdots \mathrm{,}\langle F\left(u\right)\mathrm{,}{e}_k\rangle \right)\mathrm{<mo>\; ;\; </mo>}u={\displaystyle \underset{i=1}{\overset{d}{\sum }}}{a}_i{e}_i\mathrm{.}\end{array}$

Proposition 3.1 The map G is continuous and $G\left(a\right)\cdot a$ tends to infinity when ${\Vert a\Vert }_{I{R}^k}$ tends to infinity.

Proof. Since F restricted to $V_k$ is continuous by Lemma 2.2, so G is continuous.

Let $a\in I{R}^d$ and $u={\displaystyle \underset{1\le i\le d}{\sum }{a}^i\cdot e_i}$ in $V_k$, then $G\left(a\right)\cdot a=\langle F\left(u\right)\mathrm{,}u\rangle$ and which implies that ${\Vert a\Vert }_{I{R}^d}$ tends to infinity if ${\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }$ tends to infinity.

$G\left(a\right)\cdot a={\displaystyle \underset{1\le i\le d}{\sum }}\langle F\left(u\right)\mathrm{,}{a}^i\cdot e_i\rangle =\langle F\left(u\right)\mathrm{,}u\rangle$

and

$\begin{array}{c}{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^p={\Vert {\displaystyle \underset{1\le i\le d}{\sum }}{a}^i\cdot e_i\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^p\le {\left({\displaystyle \underset{1\le i\le d}{\sum }}\left|a^i\right|\cdot {\Vert e_i\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\right)}^p\\ \le \underset{1\le i\le d}{\max }\left({\Vert e_i\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^p\right)\cdot {\left({\displaystyle \underset{1\le i\le d}{\sum }}\left|a^i\right|\right)}^p\\ \le c\cdot {\Vert a\Vert }_{I{R}^p}\mathrm{,}\end{array}$

which implies that ${\Vert a\Vert }_{I{R}^p}$ tends to infinity if ${\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }$ tends to infinity.

Now, it suffices to prove that

$\langle F\left(u\right)\mathrm{,}u\rangle \to \infty \text{when}{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\to \infty \mathrm{.}$

Indeed, thanks to the first coercivity condition and the Hölder inequality, we obtain

$\begin{array}{c}I={\displaystyle {\int }_{\Omega }}\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}Du\text{d}x\\ \ge -{\Vert {\lambda }_2\Vert }_1-{\displaystyle {\int }_{\Omega }}{\lambda }_3{\omega }_{0j}^{\alpha \mathrm{<mo>\; /\; </mo>}p}{\left|u_j\right|}^{\alpha }\text{d}x+c_2{\displaystyle \underset{1\le i\mathrm{,}j\le n\mathrm{,}m}{\sum }}{\displaystyle {\int }_{\Omega }}{\left|D_{ij}u\right|}^p{\omega }_{ij}\text{d}x\end{array}$

By the Hölder inequality, we have

$\begin{array}{c}{\displaystyle {\int }_{\Omega }}{\lambda }_3{\left|u_j\right|}^{\alpha }{\omega }_{0j}^{\alpha \mathrm{<mo>\; /\; </mo>}p}\text{d}x\le {\Vert {\lambda }_3\Vert }_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{<mo>\; /\; </mo>}\alpha {)}^{\prime }}{\left({\displaystyle {\int }_{\Omega }}{\omega }_{0j}{\left|u_j\right|}^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{<mo>\; /\; </mo>}\alpha \mathrm{<mo\; stretchy="false">\; )\; </mo>}\cdot \alpha }\right)}^{\alpha \mathrm{<mo>\; /\; </mo>}p}\\ \le c^{\prime }{\Vert {\lambda }_3\Vert }_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{<mo>\; /\; </mo>}\alpha {)}^{\prime }}{\Vert u_j\Vert }_{\mathrm{1,}p\mathrm{,}{\omega }_{0j}}\mathrm{.}\end{array}$

where $c^{\prime }$ is a constant positive. For ${\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }$ large enough, we can write

$\begin{array}{c}\left|I\right|\ge -{\Vert {\lambda }_2\Vert }_1-c^{\prime }{\Vert {\lambda }_3\Vert }_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{<mo>\; /\; </mo>}\alpha {)}^{\prime }}\cdot {\Vert u_j\Vert }_{\mathrm{1,}p\mathrm{,}{\omega }_{0j}}^{\alpha }+c_2\cdot {\displaystyle {\sum }_{1\le i\mathrm{,}j\le n\mathrm{,}m}{\Vert D{u}_j\Vert }_{\mathrm{1,}p\mathrm{,}{\omega }_{ij}}^p}\\ \ge -{\Vert {\lambda }_2\Vert }_1-c^{\prime }{\Vert {\lambda }_3\Vert }_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{<mo>\; /\; </mo>}\alpha {)}^{\prime }}\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^{\alpha }+c_2{c}^{\prime }\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^p\mathrm{.}\end{array}$

And since

$\left|I^{\prime }\right|=\left|\langle v,u\rangle \right|\le {\Vert v\Vert }_{-1,p^{\prime },{\omega }^{\ast }}\cdot {\Vert u\Vert }_{1,p,\omega }$

Finally, it follows from the growth condition ${\left(F_1\right)}^{\ast }$ and $G_1$ that:

$\begin{array}{c}\left|I^{″}\right|=\left|{\displaystyle {\int }_{\Omega }f\left(x\mathrm{,}u\mathrm{,}Du\right)\cdot u\text{d}x}\right|\\ \le \left({\Vert b_1\Vert }_{p^{\prime }}+{c^{\prime }}_1\cdot {\Vert Du\Vert }_{\mathrm{1,}p\mathrm{,}\omega }+{c^{\prime }}_2{\Vert Du\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\right)\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\\ \le c_3\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\end{array}$

$\left|I^{‴}\right|=\left|{\displaystyle {\int }_{\Omega }g\left(x\mathrm{,}u\right)\cdot Du\text{d}x}\right|\le \left({\Vert b_2\Vert }_{p^{\prime }}+{\Vert u\Vert }_{q\mathrm{,}\gamma }^{\frac{q}{p^{\prime }}}\right)\cdot {\Vert Du\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\le c_4\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\mathrm{<mo>\; ;\; </mo>}$

with $c_4$ is a constant. With; $0<\alpha <p$ and $p>1$, we get:

$\begin{array}{c}I-I^{\prime }-I^{″}\ge c_2\cdot c^{\prime }\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^p-{\Vert v\Vert }_{-\mathrm{1,}{p}^{\prime }\mathrm{,}{\omega }^{\ast }}\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }-c^{\prime }{\Vert {\lambda }_3\Vert }_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{<mo>\; /\; </mo>}\alpha {)}^{\prime }}\cdot {\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^{\alpha }\\ -{\Vert {\lambda }_2\Vert }_1-c_3\mathrm{.}{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\end{array}$ (3.1)

Consequently, by using (3.1), we deduce

$I-I^{\prime }-I^{″}\to \infty \text{as}{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\to \infty \mathrm{.}$

and

$I^{‴}\to \infty \text{as}{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\to \infty \mathrm{.}$

$\langle F\left(u\right)\mathrm{,}u\rangle \to \infty \text{as}{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\to \infty \mathrm{.}$

Remark 3.2 The properties of G allow us to construct our Galerkin approximations.

Corollary 3.1 For all $k\in IN$ , there exists $\left(u_k\right)\subset V_k$ such that $\langle F\left(u_k\right)\mathrm{,}\phi \rangle =0$ , for all $\phi \in V_k$ .

Proof By the proposition 3.1, there exists $R>0$, such that for all $a\in \partial B_R\left(0\right)\subset I{R}^d$, we have $G\left(a\right)\cdot a>0$. And the usual topological argument see [Zei 86 proposition 2.8] [17] implies that $G\left(x\right)=0$ has a solution $x\in B_R\left(0\right)$. So, for all $k\in In$, there exists $\left(u_k\right)\subset V_k$, such that

$\langle F\left(x^j{e}_j\right),e_j\rangle =0\text{for}\text{all}1\le j\le d,\text{with}d=\dim V_k$

Taking $u_k=\left(x_k^i{e}_i\right)\mathrm{,}{e}_i\in V_k$, so we obtain:

$\langle F\left(u_k\right)\mathrm{,}\phi \rangle =\mathrm{0,}\text{for}\text{all}\phi \in V_k\mathrm{.}$

Proposition 3.2 The Galerkin approximations sequence constructed in corollary (3.1) is uniformly bounded in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ ; i.e.,

There exists a constant $R>0$, such that ${\Vert u_k\Vert }_{1,p,\omega }\le R$, for all $k\in IN$.

Proof Like in the proof of proposition (3.1), we can see that

$\langle F\left(u\right)\mathrm{,}u\rangle \to \infty \text{as}{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\to \infty \mathrm{.}$

Then, there exists R satisfying $\langle F\left(u\right),u\rangle >1$ when ${\Vert u\Vert }_{1,p,\omega }>R$. Now, for the sequence of Galerkin approximations $\left(u_k\right)\subset V_k$ of corollary (3.1), which satisfying $\langle F\left(u_k\right),u_k\rangle =0$, we have the uniform bound ${\Vert u_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\le R$, for all $k\in IN$.

Remark 3.3 There exists a subsequence $\left(u_k\right)$ of the sequence $\left(u_k\right)\subset V_k$ , such that:

$u_k\rightharpoonup u$ in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$

and

$u_k\to u$ in measure in $L^r\left(\Omega \mathrm{,}I{R}^m\right)$ ;

with

$\{\begin{array}{ll}1\le r\prec \frac{nps}{n\left(s+1\right)-ps}\hfill & \text{if}ps\prec n\left(s+1\right)\hfill \\ r\ge 1\hfill & \text{if}n\left(s+1\right)\prec ps\hfill \end{array}$

The gradient sequence $\left(D{u}_k\right)$ generates the young measure ${\vartheta }_x$. Since $u_k\to u$ in measure, then $\left(u_k\mathrm{,}D{u}_k\right)$ generates the Young measure $\left({\delta }_{u\mathrm{<mo\; stretchy="false">\; (\; </mo>}x\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\otimes {\vartheta }_x\right)$, see [2]. Moreover, for almost x in $\Omega$, we have,

1) ${\vartheta }_x$ is the probability measure, i.e., ${\Vert {\vartheta }_x\Vert }_{mes}=1$.

2) ${\vartheta }_x$ is the $W^{\mathrm{1,}p\mathrm{,}\omega }$ gradient homogeneous young measure.

3) $\langle {\vartheta }_x\mathrm{,}id\rangle =Du\left(x\right)$, see [18].

Proof. See [2]. (Dolzmann, N. Humgerbuhler S. Muller, Non linear elliptic system …)

4. Passage to the Limit

Now, we are in a position to prove our main result under convenient hypotheses.

Let

$I_k=\left(\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)-\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\right)\mathrm{<mo>\; :\; </mo>}\left(D{u}_k-Du\right)\mathrm{.}$ (4.1)

Lemma 4.1 (Fatou lemma type)(See [2] ) Let: $F\mathrm{<mo>\; :\; </mo>}\Omega \times I{R}^m\times I{M}^{m\times n}\to IR$ be a Carathéodory function, and $u_k\mathrm{<mo>\; :\; </mo>}\Omega \to I{R}^m$ a measurable sequence, such that $\left(D{u}_k\right)$ generates the Young measure ${\vartheta }_x$ , with ${\Vert {\vartheta }_x\Vert }_{mes}=1$ , for a.e. $x\in \Omega$ . Then:

$\underset{k\to \infty }{\lim \inf }{\displaystyle {\int }_{\Omega }}F\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\text{d}x\ge {\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}F\left(x\mathrm{,}u\mathrm{,}\zeta \right)\text{d}{\vartheta }_x\left(\zeta \right)\text{d}x\mathrm{,}$ (4.2)

which provided that the negative part of $F\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)$ is equi-integrable.

Proof.

Lemma 4.2 Let $p>1$ and $u_k$ be a sequence which is uniformly bounded in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ . There exists a subsequence of $u_k$ (for convenience not relabeled) and a function $u\in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ such that $u_k\rightharpoonup u$ in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$

And such that $u_k\to u$ in measure on $\Omega$ and in $L^r\left(\Omega \mathrm{,}I{R}^m\right)$ , with:

$\{\begin{array}{ll}1\le r\prec \frac{nps}{n\left(s+1\right)-ps}\hfill & \text{if}ps\prec n\left(s+1\right)\hfill \\ r\ge 1\hfill & \text{if}n\left(s+1\right)\prec ps\hfill \end{array}$

Proof. see [10].

Lemma 4.3 The sequence $\left(I_k\right)$ is equi-integrable.

Proof

We have

$\begin{array}{c}{I}_k=\left(\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)-\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\right)\mathrm{<mo>\; :\; </mo>}\left(D{u}_k-Du\right)\\ =\left[\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D{u}_k\right]-\left[\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}Du\right]\\ -\left[\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}D{u}_k\right]+\left[\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}Du\right]\\ =I_k^1+I_k^2+I_k^3+I_k^4\end{array}$ (4.3)

We denote ${\left(I_k^1\right)}^{-}=-{\left[\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D{u}_k\right]}^{-}$. Thanks to the coercivity condition (H₂), we have

$\begin{array}{c}{\displaystyle {\int }_{{\Omega }^{\prime }}}\left|{\left(I_k^1\right)}^{-}\right|\text{d}x\le {\displaystyle {\int }_{\Omega }}\left|{\lambda }_2\right|+c_2{\displaystyle \underset{1\le j\le m}{\sum }}{\omega }_{0j}^{\frac{\alpha }{p}}\left|{\lambda }_3\right|\cdot {\left|u_{kj}\right|}^{\alpha }+c{\displaystyle \underset{1\le i\mathrm{,}j\le n\mathrm{,}m}{\sum }}{\omega }_{ij}{\left|D_{ij}{u}_k\right|}^p\text{d}x\\ \le {\Vert {\lambda }_2\Vert }_1+{\displaystyle {\int }_{{\Omega }^{\prime }}}{\left({\displaystyle \underset{1\le j\le m}{\sum }}{\omega }_{0j}^{\alpha \mathrm{<mo>\; /\; </mo>}p}{\left|u_{kj}\right|}^{\alpha }\right)}^{p\mathrm{<mo>\; /\; </mo>}\alpha }{\Vert {\lambda }_3\Vert }_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{<mo>\; /\; </mo>}\alpha {)}^{\prime }}+c_2{\Vert u_k\Vert }_{\mathrm{1,}\omega \mathrm{,}p}^p\end{array}$ (4.4)

with $p/\alpha \ge 1$. Therefore,

$\begin{array}{c}{\displaystyle {\int }_{{\Omega }^{\prime }}}\left|{\left(I_k^1\right)}^{-}\right|\text{d}x\le {\Vert {\lambda }_2\Vert }_1+{\left({\displaystyle \underset{1\le j\le m}{\sum }}{\omega }_{0j}{\left|u_{kj}\right|}^p\right)}^{\alpha /p}{\Vert {\lambda }_3\Vert }_{(p/\alpha {)}^{\prime }}+c_2{\Vert u_k\Vert }_{1,\omega ,p}^p\\ \le {\Vert {\lambda }_2\Vert }_1+{\Vert u_k\Vert }_{p,{\bar{\omega }}_{00}}^{\alpha }{\Vert {\lambda }_3\Vert }_{(p/\alpha {)}^{\prime }}+c_2{\Vert u_k\Vert }_{1,\omega ,p}^p\\ <\infty ,\end{array}$

for all ${\Omega }^{\prime }\subset \Omega$.

Similarly for $\left|{\left(I_k^4\right)}^{-}\right|$.

Now, by using the growth condition (H₂) and the Hardy inequality (H₀), we have

$\begin{array}{l}{\displaystyle {\int }_{{\Omega }^{\prime }}}\left|{\left(I_k^2\right)}^{-}\right|\text{d}x={\displaystyle {\int }_{{\Omega }^{\prime }}}\left|\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D{u}_k\right|\text{d}x\\ \le \beta {\displaystyle {\int }_{{\Omega }^{\prime }}}{\omega }_{rs}^{\mathrm{1\; <mo>\; /\; </mo>}p}\left({\lambda }_1+c_1{\displaystyle \underset{1\le j\le m}{\sum }}{\gamma }_j^{\mathrm{1\; <mo>\; /\; </mo>}{p}^{\prime }}\cdot {\left|u_{kj}\right|}^{q\mathrm{<mo>\; /\; </mo>}{p}^{\prime }}+c_2{\displaystyle \underset{1\le i\mathrm{,}j\le n\mathrm{,}m}{\sum }}{\omega }_{ij}^{\mathrm{1\; <mo>\; /\; </mo>}{p}^{\prime }}{\left|D_{ij}{u}_k\right|}^{p-1}\right)D_{rs}{u}_k\text{d}x\mathrm{.}\end{array}$ (4.5)

Thus, by the Hölder inequality, we obtain

$\begin{array}{c}{\displaystyle {\int }_{{\Omega }^{\prime }}}\left|{\left(I_k^2\right)}^{-}\right|\text{d}x\le \beta [{\Vert {\lambda }_1\Vert }_{p^{\prime }}{\left({\displaystyle {\int }_{{\Omega }^{\prime }}}{\left|D_{rs}{u}_k\right|}^p{\omega }_{rs}\text{d}x\right)}^{1/p}\\ +c_1{\left({\displaystyle {\int }_{{\Omega }^{\prime }}}{\left|D_{rs}{u}_k\right|}^p{\omega }_{rs}\text{d}x\right)}^{1/p}{\left({\displaystyle {\int }_{{\Omega }^{\prime }}}{\left({\displaystyle {\sum }_{1\le j\le m}}{\gamma }_j^{1/p^{\prime }}{\left|u_{kj}\right|}^{q/p^{\prime }}\right)}^{p^{\prime }}dx\right)}^{1/p^{\prime }}\\ +c_1\left({\displaystyle \underset{1\le j\le m}{\sum }}{\displaystyle {\int }_{{\Omega }^{\prime }}}{\left({\left|D_{ij}{u}_k\left(x\right)\right|}^{p^{\prime }(p-1)}{\omega }_{ij}\text{d}x\right)}^{1/p^{\prime }}\right){\left({\displaystyle {\int }_{{\Omega }^{\prime }}}{\left|D_{rs}{u}_k\right|}^p{\omega }_{rs}\text{d}x\right)}^{1/p}].\end{array}$ (4.6)

So, by combining (4.5) and (4.6), we deduce that

${\displaystyle {\int }_{{\Omega }^{\prime }}}\left|\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D{u}_k\right|\text{d}x\le c^{\prime }\beta \left({\Vert {\lambda }_1\Vert }_{p^{\prime }}{\Vert u_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }+{\Vert u_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\right)<\infty \mathrm{.}$ (4.7)

Similarly to $\left|{\left(I_k^2\right)}^{-}\right|$, we obtain $\left|{\left(I_k^3\right)}^{-}\right|$. Finally: $I_k$ is equi-integrable.

We choose a sequence ${\phi }_k$ such that ${\phi }_k$ belongs to the same space $V_k$ and ${\phi }_k\to \phi$ in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$, this allows us in particular, to use $u_k-{\phi }_k$ as a test function in (3.1). We have:

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}\left|\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}\left(D{u}_k-D{\phi }_k\right)\right|\text{d}x\\ =\langle v\mathrm{,}{u}_k-{\phi }_k\rangle +{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\cdot \left(u_k-{\phi }_k\right)\text{d}x\\ -{\displaystyle {\int }_{\Omega }}g\left(x\mathrm{,}{u}_k\right)\mathrm{<mo>\; :\; </mo>}\left(D{u}_k-D{\phi }_k\right)\text{d}x\mathrm{.}\end{array}$ (4.8)

The first term on the right in 4.8 converge to zero since $\left(u_k-{\phi }_k\right)\rightharpoonup 0$ in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$. By the choice of ${\phi }_k$, the sequence ${\phi }_k$ uniformly bounded in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$, and lemma (4.2). Next, for the second term: $I{I}_k={\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\cdot \left(u_k-{\phi }_k\right)\text{d}x$ in 4.8 it follows from the growth condition $F_1^{\ast }$ and the Hölder inequality that:

$\begin{array}{c}\left|I{I}_k\right|\le \left({\Vert b_1\Vert }_{p^{\prime }}+{c^{\prime }}_1\cdot {\Vert D\left(u_k-{\phi }_k\right)\Vert }_{\mathrm{1,}p\mathrm{,}\omega }+{c^{\prime }}_2\cdot {\Vert D\left(u_k-{\phi }_k\right)\Vert }^{\frac{p}{p^{\prime }}}\right)\cdot {\Vert u_k-{\phi }_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\\ \le \left({\Vert b_1\Vert }_{p^{\prime }}+c\cdot {\Vert D\left(u_k-{\phi }_k\right)\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\right)\cdot {\Vert u_k-{\phi }_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\end{array}$

By the equivalence of the norm in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ and the sequence $u_k$ is uniformly bounded in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$, ${\Vert u_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }$ is bounded.

Moreover, by the construction of ${\phi }_k$, and lemma (4.2) we have:

${\Vert u_k-{\phi }_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\le {\Vert u_k-u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }+{\Vert u-{\phi }_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }$

$\left({\Vert u_k-u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }+{\Vert u-{\phi }_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\right)\to 0$

We infer that the second term in 4.8 vanishes as $k\to \infty$. Finally, for the last term

$II{I}_k={\displaystyle {\int }_{\Omega }}g\left(x,u_k\right):D\left(u_k-{\phi }_k\right)\text{d}x$

in 4.8, we note that

$g\left(x\mathrm{,}{u}_k\right)\to g\left(x\mathrm{,}u\right)$

Strongly in $L^{p^{\prime }}\left(\Omega \mathrm{,}{M}^{m\times n}\right)$ by $\left(G_0\right)$, $\left(G_1\right)$ and lemma (4.2).

Indeed we may assure that $u_k\to u$ almost everywhere.

$\begin{array}{c}II{I}_k\le \left({\Vert b_2\Vert }_{p^{\prime }}+{\Vert u_k-{\phi }_k\Vert }_{q\mathrm{,}\gamma }^{\frac{q}{p^{\prime }}}\right)\cdot {\Vert D\left(u_k-{\phi }_k\right)\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\\ \le c^{\prime }\cdot \left({\Vert b_2\Vert }_{p^{\prime }}+{\Vert u_k-{\phi }_k\Vert }_{q\mathrm{,}\gamma }^{\frac{q}{p^{\prime }}}\right)\cdot {\Vert \left(u_k-{\phi }_k\right)\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\\ \le c^{\prime }\cdot \left({\Vert b_2\Vert }_{p^{\prime }}+{\Vert u_k-{\phi }_k\Vert }_{q\mathrm{,}\gamma }^{\frac{q}{p^{\prime }}}\right)\cdot \left({\Vert u_k-u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }+{\Vert {\phi }_k-u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\right)\end{array}$

${\Vert {\phi }_k-u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\to 0$, ${\Vert u_k-u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\to 0$ and ${\Vert u_k-{\phi }_k\Vert }_{q\mathrm{,}\gamma }^{\frac{q}{p^{\prime }}}\to 0$

Now, we consider ${\left(I_k\right)}^{\prime }=\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}\left(D{u}_k-Du\right)$. We have, ${I^{\prime }}_k$ is equi-integrable because $I_k$ it is. So, we define

$\begin{array}{c}X=\lim \inf {\displaystyle {\int }_{\Omega }}{I}_k\text{d}x=\lim \inf {\displaystyle {\int }_{\Omega }}{\left(I_k\right)}^{\prime }\text{d}x\\ \ge {\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\left(\sigma \left(x,u,\lambda \right):\left(\lambda -Du\right)\right)\text{d}{\vartheta }_x(\; \lambda \; )\end{array}$

So to prove (??), it suffices to prove that:

$X\le 0.$ (4.9)

Let $\epsilon >0$, so there exists $k_0\in IN$ such that, for all $k>k_0$, we have $dist\left(u,V_k\right)<\epsilon$ since: $\lim {\inf }_{{\phi }_k\in V_k}{\Vert u-{\phi }_k\Vert }_{1,p,\omega }<\epsilon$, $\left(u_k\rightharpoonup u\right)$

Or in an equivalent manner $dist\left(u_k-u,V_k\right)<\epsilon$, $\forall k>k_0$ then for all $v_k\in V_k$, we have

$\begin{array}{l}X=\underset{k\to \infty }{\lim \inf }{\displaystyle {\int }_{\Omega }}\left(\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}\left(D{u}_k-Du\right)\right)\text{d}x\\ =\underset{k\to \infty }{\lim \inf }\left[{\displaystyle {\int }_{\Omega }}\left(\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\left(u_k-u-{\phi }_k\right)\right)\text{d}x+{\displaystyle {\int }_{\Omega }}\left(\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\left({\phi }_k\right)\right)\right]\end{array}$

Combining (H₂) and (0.1), we get

$\begin{array}{l}X\le \lim \underset{k\to \infty }{\inf }{\displaystyle {\int }_{\Omega }}\beta {\omega }_{rs}^{1/p}\left[{\lambda }_1+c_1{\displaystyle \underset{1\le j\le m}{\sum }}{\gamma }_j^{1/p^{\prime }}{\left|u_{k_j}\right|}^{q/p^{\prime }}+c_1{\displaystyle \underset{1\le i,j\le n,m}{\sum }}{\omega }_{ij}^{1/p^{\prime }}{\left|D_{ij}{u}_k\right|}^{p-1}\right]\\ \times \left|D_{rs}\left(u_k-u-{\phi }_k\right)\right|\text{d}x+\langle v\mathrm{,}{\phi }_k\rangle \mathrm{.}\end{array}$

For all $\epsilon >0$, we choose ${\phi }_k\in V_k$ such that

${\Vert u_k-u-{\phi }_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\le 2\epsilon \mathrm{,}$ (4.10)

For all $k\ge k_0$, which implies that

$\left|\langle v\mathrm{,}{\phi }_k\rangle \right|\le \left|\langle v\mathrm{,}{\phi }_k+\left(u-u_k\right)\rangle \right|+\left|\langle v\mathrm{,}{u}_k-u\rangle \right|\le 2\epsilon {\Vert v\Vert }_{-\mathrm{1,}{p}^{\prime }\mathrm{,}{\omega }^{\ast }}+o(\; k\; )$

Hence ${\lim }_{k\to \infty }\langle v\mathrm{,}{u}_k-u\rangle =0$. According to Hölder and Hardy inequalities, and by (4.1) we deduce that

$\begin{array}{c}X\le \underset{k\to \infty }{\lim \inf }c\beta ({\Vert {\lambda }_1\Vert }_{p^{\prime }}{\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\left(u_k-u-{\phi }_k\right)\right|}^p\cdot {\omega }_{rs}\text{d}x\right)}^{1/p}\\ +c_1{\left({\displaystyle {\int }_{\Omega }}{\left|u_k\right|}^q\cdot \gamma \right)}^{1/p^{\prime }}\cdot {\left({\displaystyle {\int }_{\Omega }}{\left|D_{rs}\left(u_k-u-{\phi }_k\right)\right|}^p{\omega }_{rs}\text{d}x\right)}^{1/p}\\ +c_1{\left({\displaystyle \sum }{\displaystyle {\int }_{\Omega }}{\omega }_{ij}{\left|D_{ij}u\right|}^{p^{\prime }(p-1)}\right)}^{1/p^{\prime }}\cdot {\left({\displaystyle {\int }_{\Omega }}{\omega }_{rs}{\left|D_{rs}\left(u_k-u-{\phi }_k\right)\right|}^p\right)}^{1/p})+\left|\langle v,{\phi }_k\rangle \right|\\ \le \underset{k\to \infty }{\lim \inf }c\left({\Vert {\lambda }_1\Vert }_{p^{\prime }}\cdot {\Vert u_k-u-{\phi }_k\Vert }_{1,p,\omega }\right)+{\Vert u_k\Vert }_{1,p,\omega }^q{\Vert u_k-u-{\phi }_k\Vert }_{1,p,\omega }\\ +2\epsilon {\Vert v\Vert }_{-1,p^{\prime },{\omega }^{\ast }}+o(\; k\; )\end{array}$

Therefore,

$X\le 2\epsilon c\beta \left({\Vert {\lambda }_1\Vert }_{p^{\prime }}+{\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^q+{\Vert v\Vert }_{-\mathrm{1,}{p}^{\prime }\mathrm{,}\omega \mathrm{<mo>\; *\; </mo>}}\right)\mathrm{.}$

which proves that $X\le 0$, and finally

${\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}\lambda \text{d}{\vartheta }_x\text{d}x\le {\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}Du\text{d}{\vartheta }_x\left(\lambda \right)\text{d}x\mathrm{.}$

Proof of theorem:

For arbitrary $\phi$ in $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$. It follows from the continuity condition $\left(F_0^{\ast }\right)$ and $\left(G_0\right)$ that

$f\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\cdot \phi \left(x\right)\to f\left(x\mathrm{,}u\mathrm{,}Du\right)\cdot \phi (\; x\; )$

and

$g\left(x\mathrm{,}{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \left(x\right)\to g\left(x\mathrm{,}u\right)\mathrm{<mo>\; :\; </mo>}D\phi (\; x\; )$

almost everywhere. Since, by the growth conditions $\left(F_1^{\ast }\right)$, $\left(G_1\right)$ and the uniform bound of $u_k$, $f\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\cdot \phi \left(x\right)$ and $g\left(x\mathrm{,}{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \left(x\right)$ are equi-integrable, it follows that the Vitali’s theorem. This implies that:

$\underset{k\to \infty }{\lim }{\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\cdot \phi \left(x\right)\text{d}x={\displaystyle {\int }_{\Omega }}f\left(x\mathrm{,}u\mathrm{,}Du\right)\cdot \phi \left(x\right)\text{d}x$

for all $\phi \in {\displaystyle {\cup }_{k=1}^{\infty }{V}_k}$ and

$\underset{k\to \infty }{\lim }{\displaystyle {\int }_{\Omega }}g\left(x\mathrm{,}{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \left(x\right)\text{d}x={\displaystyle {\int }_{\Omega }}g\left(x\mathrm{,}u\right)\mathrm{<mo>\; :\; </mo>}D\phi \left(x\right)\text{d}x$

for all $\phi \in {\displaystyle {\cup }_{k=1}^{\infty }{V}_k}$ We will start with the easiest case

(d): $F\mapsto \sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is strict p-quasi-monotone. (4.11)

Indeed, we assume that ${\vartheta }_x$ is not a Dirac mass on the set M with $x\in M$ of positive Lebesgue measure $\left|M\right|>0$. Moreover, by the strict p-quasi-monotonicity of $\sigma \left(x\mathrm{,}u\mathrm{,}\cdot \right)$ and ${\vartheta }_x$ is an homogeneous $W^{\mathrm{1,}p}$ gradient young measure for a.e. $x\in M$. So, for a.e. $x\in M$, with $\bar{\lambda }=\langle {\vartheta }_x,Id\rangle =apDu\left(x\right)$, with $apDu\left(x\right)$ is the differentiable approximation in x. We get

$\begin{array}{l}{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x,u,\lambda \right):\left(\lambda -Du\right)\text{d}{\vartheta }_x\left(\lambda \right)\\ >{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x,u,Du\right):\left(\lambda -Du\right)\text{d}{\vartheta }_x\left(\lambda \right)\\ >\sigma \left(x,u,Du\right):{\displaystyle {\int }_{I{M}^{m\times n}}}\lambda \text{d}{\vartheta }_x\left(\lambda \right)-\sigma \left(x,u,Du\right):Du{\displaystyle {\int }_{I{M}^{m\times n}}}\text{d}{\vartheta }_x\left(\lambda \right)\\ >\left(\sigma \left(x,u,Du\right):Du-\sigma \left(x,u,Du\right):Du\right)=0\\ >0\end{array}$

On the other hand (4.9), integrating over $\Omega$, and using the div-cul inequality we have:

$\begin{array}{l}{\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}\lambda \text{d}{\vartheta }_x\left(\lambda \right)\text{d}x\\ >{\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}Du\text{d}{\vartheta }_x\left(\lambda \right)\text{d}x\\ \ge {\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}\lambda \text{d}{\vartheta }_x\left(\lambda \right)\text{d}x\mathrm{.}\end{array}$

Which is a contradiction with (3.8). Thus ${\vartheta }_x={\delta }_{\bar{\lambda }}={\delta }_{Du(x)}$ for a.e. $x\in \Omega$. Therefore, $D{u}_k\to Du$ in measure when k tends to infinity. Then, we get $\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\to \sigma \left(x\mathrm{,}u\mathrm{,}Du\right)$ for all $x\in \Omega$. In the other hand, for all $\phi \in {\displaystyle \underset{k\in IN}{\cup }}{\vartheta }_k$ ; $\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \to \sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}D\phi$ a.e. $x\in \Omega$. Moveover, for all ${\Omega }^{\prime }\subset \Omega$ measurable, it is easy to see that:

${\displaystyle {\int }_{{\Omega }^{\prime }}}\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{d}x\le c\beta \left({\Vert {\lambda }_1\Vert }_{p^{\prime }}+{\Vert u_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^{q\mathrm{<mo>\; /\; </mo>}{p}^{\prime }}+{\Vert u_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }^{p\mathrm{<mo>\; /\; </mo>}{p}^{\prime }}\right){\Vert u\Vert }_{\mathrm{1,}p\mathrm{,}\omega }<\infty \mathrm{,}$

because ${\Vert u_k\Vert }_{\mathrm{1,}p\mathrm{,}\omega }\le R$. And thanks to Vitali’s theorem, we obtain:

$\langle F\left(u\right)\mathrm{,}\phi \rangle =0$, for all $\phi \in {\displaystyle \underset{k\in IN}{\cup }}{\vartheta }_k$.

which proves the theorem in this case.

Remark 4.1 Before treating the cases (a),(b) and (c) of (H₃), we note that

${\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\left(\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)-\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du\right)\text{d}{\vartheta }_x\left(\lambda \right)\text{d}x\le 0$ (4.12)

Since

${\displaystyle {\int }_{\Omega }}{\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du\right)\text{d}{\vartheta }_x\left(\lambda \right)\text{d}x=\mathrm{0,}$

thanks to the div-Curl inequality in (4.9). On the other hand, the integrand in (4.12) is non negative, by the monotonicity of $\sigma$. Consequently, the integrating should be null, a.e., with respect to the product measure $\text{d}{\vartheta }_x\otimes \text{d}x$, which mean

$\left(\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)-\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du\right)=0\text{in}spt{\vartheta }_x\mathrm{.}$ (4.13)

Thus,

$spt{\vartheta }_x\subset \left\{\lambda \in I{M}^{m\times n}\mathrm{<mo>\; /\; </mo>}\left(\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)-\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du\right)=0\right\}\mathrm{.}$ (4.14)

Case c: We prove that, the map $F\mapsto \sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is strictly monotone, for all $x\in \Omega$ and for all $u\in I{R}^m$.

Sine $\sigma$ is strict monotone, and according to (4.14),

$spt{\vartheta }_x=\left\{Du\right\},\text{i}\text{.e},{\vartheta }_x={\delta }_{Du},\text{a}\text{.e}\text{.}\text{in}\Omega ,$

which implies that, $D{u}_k\to Du$ in measure. For the rest of our prove is similarly to case d.

Case b: We start by showing that for almost all $x\in \Omega$, the support of ${\vartheta }_x$ is contained in the set where W agrees with the supporting hyper-plane.

$L=\left\{\left(\lambda \mathrm{,}W\left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)+\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -\bar{\lambda }\right)\right)\right\}\text{with}\bar{\lambda }=Du\left(x\right)\mathrm{.}$

So, it suffices to prove that

$spt{\vartheta }_x\subset K_x=\left\{\lambda \in I{M}^{m\times n}\mathrm{<mo>\; /\; </mo>}W\left(x\mathrm{,}u\mathrm{,}\lambda \right)=W\left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)+\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -\bar{\lambda }\right)\right\}$ (4.15)

If $\lambda \in spt{\vartheta }_x$, thanks to (4.14), we have

$\left(1-t\right)\cdot \left(\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)-\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\right)\mathrm{<mo>\; :\; </mo>}\left(Du-\lambda \right)=\mathrm{0,}\text{for}\text{all}t\in \left[\mathrm{0,1}\right]\mathrm{.}$ (4.16)

On the other hand, since $\sigma$ is monotone, for all $t\in \left[\mathrm{0,1}\right]$ we have:

$\left(1-t\right)\cdot \left(\sigma \left(x\mathrm{,}u\mathrm{,}Du+t\cdot \left(\lambda -Du\right)\right)-\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\right)\mathrm{<mo>\; :\; </mo>}\left(Du-\lambda \right)\ge 0.$ (4.17)

By subtracting (4.16) from (4.17), we get

$\left(1-t\right)\left[\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }+t\left(\lambda -\bar{\lambda }\right)\right)-\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\right]\mathrm{<mo>\; :\; </mo>}\left(\bar{\lambda }-\lambda \right)\ge \mathrm{0,}$ (4.18)

for all $t\in \left[\mathrm{0,1}\right]$. Doing the same by the monotonicity in (4.18), we obtain

$\left(1-t\right)\left[\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }+t\left(\lambda -\bar{\lambda }\right)\right)-\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\right]\mathrm{<mo>\; :\; </mo>}\left(\bar{\lambda }-\lambda \right)\le 0.$ (4.19)

Combining (4.18) and (4.19), we conclude that

$\left(1-t\right)\left[\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }+t\left(\lambda -\bar{\lambda }\right)\right)-\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\right]\mathrm{<mo>\; :\; </mo>}\left(\bar{\lambda }-\lambda \right)=\mathrm{0,}$ (4.20)

for all $t\in \left[\mathrm{0,1}\right]$, and for all $\lambda \in spt{\vartheta }_x$.

Now, it follows from (4.19) that

$\begin{array}{c}W\left(x\mathrm{,}u\mathrm{,}\lambda \right)=W\left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)+\left(W\left(x\mathrm{,}u\mathrm{,}\lambda \right)-W\left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\right)\\ =W\left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)+{\displaystyle {\int }_0^1}\left[\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)+t\left(\lambda -\bar{\lambda }\right)\right]\mathrm{<mo>\; :\; </mo>}\left(\lambda -\bar{\lambda }\right)\text{d}t\\ =W\left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)+\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -\bar{\lambda }\right)\end{array}$

Witch prove (4.15).

Now, by the coercivity of W, we get

$W\left(x\mathrm{,}u\mathrm{,}\lambda \right)\ge W\left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)+\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -\bar{\lambda }\right)\mathrm{,}$

for all $\lambda \in I{M}^{m\times n}$. Therefore,

L is a supporting hyper-plane, for all

$\lambda \in K_x$. (4.21)

Moveover, the mapping $\lambda \mapsto W\left(x\mathrm{,}u\mathrm{,}\lambda \right)$ is continuously differentiable, so we obtain

$\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)=\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\mathrm{,}\text{for}\text{all}\lambda \in K_x\mathrm{.}$ (4.22)

Thus,

$\bar{\sigma }\left(x\right)={\displaystyle {\int }_{I{M}^{m\times n}}}\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\text{d}{\vartheta }_x\left(\lambda \right)=\sigma \left(x\mathrm{,}u\mathrm{,}\bar{\lambda }\right)\mathrm{.}$ (4.23)

Now, we consider the Carathéodory function

$g_v\left(x\mathrm{,}u\mathrm{,}\rho \right)=\left|\left(\sigma \left(x\mathrm{,}u\mathrm{,}\rho \right)-\bar{\sigma }\left(x\right)\right)\mathrm{<mo>\; :\; </mo>}D\phi \right|\mathrm{,}$

and lets $g_k\left(x\right)=g_{\phi }\left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)$ is equi-integrable. Thus, thanks to BALL’s theorem, see [6] $g_k\rightharpoonup g$ weakly in $L^1\left(\Omega \right)$, and the weakly limit of g is given by

$\begin{array}{c}{\bar{g}}_{\phi }\left(x\right)={\displaystyle \int }{\displaystyle {\int }_{I{R}^m\times I{M}^{m\times n}}}\left|\sigma \left(x\mathrm{,}\eta \mathrm{,}\lambda \right)-\bar{\sigma }\left(x\right)\right|\text{d}{\delta }_{u\left(x\right)}\left(\eta \right)\otimes \text{d}{\vartheta }_x\left(\lambda \right)\\ ={\displaystyle {\int }_{spt{\vartheta }_x}}\left|\sigma \left(x\mathrm{,}u\left(x\right)\mathrm{,}\lambda \right)-\bar{\sigma }\left(x\right)\right|\text{d}{\vartheta }_x\left(\lambda \right)\\ =0.\end{array}$

According to (4.22) and (4.23), and since $g_k\ge 0$, it follow that $g_k\to 0$ strongly in $L^1\left(\Omega \right)$ by Fatou lemma, which gives

$\underset{k\to \infty }{\lim }{\displaystyle {\int }_{\Omega }}\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{d}x={\displaystyle {\int }_{\Omega }}\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{d}x\mathrm{.}$

Thus

$\langle F\left(u\right)\mathrm{,}\phi \rangle =\mathrm{0,}\forall \phi \in {\displaystyle \underset{k\in IN}{\cup }}{V}_k\mathrm{.}$

This completes the proof of the case (b).

Case (a): In this case, on $spt{\vartheta }_x$, we affirm that,

$\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}M=\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}M+\left({\nabla }_F\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}M\right)\mathrm{<mo>\; :\; </mo>}\left(Du-\lambda \right)\mathrm{,}$ (4.24)

for all $M\in I{M}^{m\times n}$, where ${\nabla }_F$ is the derivative with respect to the third variable of $\sigma$ and $\bar{\lambda }=Du\left(x\right)$.

Thanks to the monotonicity of $\sigma$, we have

$\left(\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)-\sigma \left(x\mathrm{,}u\mathrm{,}Du+tM\right)\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du-tM\right)\ge \mathrm{0,}\text{for}\text{all}t\in IR\mathrm{.}$

By invoking (4.19), we obtain

$\begin{array}{l}-\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}\left(tM\right)\\ \ge -\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du\right)+\sigma \left(x\mathrm{,}u\mathrm{,}Du+tM\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du-t\cdot M\right)\mathrm{.}\end{array}$

On the other hand, $F\mapsto \sigma \left(x\mathrm{,}u\mathrm{,}F\right)$ is a $C^1$ function, so

$\sigma \left(x\mathrm{,}u\mathrm{,}Du+tM\right)=\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)+{\nabla }_F\left(x\mathrm{,}u\mathrm{,}Du\right)\cdot \left(tM\right)+o\left(t\right)\mathrm{.}$

Thus

$\begin{array}{l}-\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}\left(t\cdot M\right)\\ \ge -\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}\left(tM\right)+{\nabla }_F\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\left(t\cdot M\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du\right)+o\left(t\right)\mathrm{,}\end{array}$

which gives

$\begin{array}{l}-\sigma \left(x\mathrm{,}u\mathrm{,}\lambda \right)\mathrm{<mo>\; :\; </mo>}\left(t\cdot M\right)\\ \ge t\left[{\nabla }_F\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}\left(M\right)\mathrm{<mo>\; :\; </mo>}\left(\lambda -Du\right)-\sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}\left(M\right)\right]+o\left(t\right)\mathrm{,}\end{array}$

t is arbitrary in (4.24).

Finally for all $\phi \in {\displaystyle {\cup }_{k\in IN}}{V}_k$ the sequence $\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi$ is equi-integrable. Then, by the BALL’s theorem, see [1] the weak limit is ${\displaystyle {\int }_{spt{\vartheta }_x}}\sigma \left(x,u,\lambda \right):D\phi \text{d}{\vartheta }_x(\; \lambda \; )$

By choosing $M=Du$ in (4.24), we obtain

$\begin{array}{l}{\displaystyle {\int }_{spt{\vartheta }_x}}\left(Du-\lambda \right)\left(\sigma \left(x,u,\lambda \right):D\phi \right):D\phi \text{d}{\vartheta }_x\left(\lambda \right)\\ ={\displaystyle {\int }_{spt{\vartheta }_x}}\sigma \left(x,u,Du\right):D\phi \text{d}{\vartheta }_x\left(\lambda \right)+{\left({\nabla }_F\sigma \left(x,u,Du\right):D\phi \right)}^t{\displaystyle {\int }_{spt{\vartheta }_x}}\left(Du-\lambda \right)\text{d}{\vartheta }_x\left(\lambda \right)\\ =\left(\sigma \left(x,u,Du\right):D\phi \right){\displaystyle {\int }_{spt{\vartheta }_x}}\text{d}{\vartheta }_x\left(\lambda \right)=\sigma \left(x,u,Du\right):D\phi .\end{array}$

Hence:

$\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \to \sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}D\phi \text{strongly}$

This proves that

$\langle F\left(u\right)\mathrm{,}\phi \rangle =0\text{for}\text{all}\phi \in {\displaystyle \cup }{V}_k\mathrm{.}$

And since ${\displaystyle {\cup }_{}}{V}_k$ is dense in $W_0^{\mathrm{1,}P}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$, so u is a weak solution of ${\left(QES\right)}_{f\mathrm{,}g}^{\ast }$, as desired.

Remark 4.2 In case (b) $\sigma \left(x\mathrm{,}{u}_k\mathrm{,}D{u}_k\right)\mathrm{<mo>\; :\; </mo>}D\phi \to \sigma \left(x\mathrm{,}u\mathrm{,}Du\right)\mathrm{<mo>\; :\; </mo>}D\phi$ strongly, but in the case (c) and (d) $D{u}_k\to Du$ in measure.

Exemple 4.1 We shall suppose that the weight functions satisfy: $w_{i_{0}j}=0,j=1,2,\cdots ,m$ for some $i_0\in I^c$ ; and ${\omega }_{ij}\left(x\right)=w\left(x\right)\mathrm{<mo>\; ;\; </mo>}x\in \Omega$ , with $I^c\cup I=\left\{\mathrm{0\; <mo>\; ;\; </mo>1\; <mo>\; ;\; </mo>2\; <mo>\; ;\; </mo>}\cdots \mathrm{<mo>\; ;\; </mo>}n\right\}$ , for all $i\in I\bigsqcup I^c$ , $j=1,2,\cdots ,m$, and $i\ne i_0$ with $w\left(x\right)>0$ a.e in $\Omega$ then, we can consider the Hardy inequality in the form:

${\left({\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|u_j\left(x\right)\right|}^q{\gamma }_j\left(x\right)\text{d}x\right)}^{\frac{1}{q}}\le c{\left({\displaystyle \underset{1\le i\le N,1\le j\le m}{\sum }}{\displaystyle {\int }_{\Omega }}{\left|D_{ij}u\right|}^p{\omega }_{ij}\right)}^{\frac{1}{p}},$

for every $u\in W_0^{\mathrm{1,}p}\left(\Omega \mathrm{,}\omega \mathrm{,}I{R}^m\right)$ with a constant $c>0$ independent of u and for some $q>p^{\prime }$. Let us consider the Carathéodory functions: $(\; \star \; )$

${\sigma }_{ij}\left(x,\eta ,{\xi }_I\right)=\omega \left(x\right){\left|{\xi }_{ij}\right|}^{p-1}sng\left({\xi }_{ij}\right),j=1,2,\cdots ,m,i\in I$

${\sigma }_{ij}\left(x,\eta ,{\xi }_{I^c}\right)=\omega \left(x\right){\left|{\xi }_{ij}\right|}^{p-1}sng\left({\xi }_{ij}\right),j=1,2,\cdots ,m,i\in I^c,i\ne i_0$

${\sigma }_{i_{0}j}\left(x,\eta ,{\xi }_{I^c}\right)=0,j=1,2,\cdots ,m$

$f_j\left(x\mathrm{,}\eta \mathrm{,}\xi \right)=-sign\left(\xi \right){\displaystyle \underset{rs}{\sum }}{\omega }_{rs}^{\frac{1}{p^{\prime }}}{\left|\xi \right|}^{p-1}{\omega }_{0j}^{\frac{1}{p}}$

The above functions defined by $(\star )$ satisfies the growth conditions (H₂).

In particular, let use the special weight function $\omega ,\gamma$ expressed in term of the distance to the boundary $\partial \Omega$ denote $d\left(x\right)=dist\left(x\mathrm{<mo>\; ;\; </mo>}\partial \Omega \right)$ and $\omega \left(x\right)=d^{\lambda }\left(x\right)$, ${\gamma }_j\left(x\right)=d^{\mu }\left(x\right)$ the hardy inequality reads:

${\left({\displaystyle \underset{j=1}{\overset{m}{\sum }}}{\displaystyle {\int }_{\Omega }}{\left|u_j\left(x\right)\right|}^q{\text{d}}^{\mu }\left(x\right)\text{d}x\right)}^{\frac{1}{q}}\le c{\left({\displaystyle \underset{1\le i\le N,1\le j\le m}{\sum }}{\displaystyle {\int }_{\Omega }}{\left|D_{ij}u\right|}^p{\text{d}}^{\lambda }\left(x\right)\right)}^{\frac{1}{p}},$

and the corresponding $W_0^{\mathrm{1,}p}\left(\Omega \mathrm{<mo>\; ;\; </mo>}\omega \mathrm{<mo>\; ;\; </mo>}{R}^m\right)$ ↪ $L^q\left(\Omega \mathrm{<mo>\; ;\; </mo>}\gamma \mathrm{<mo>\; ;\; </mo>}{R}^m\right)$ is compact if:

1) For, $1<p\le q<\infty$

$\lambda <p-1;\frac{n}{q}-\frac{n}{p}+1\ge 0;\frac{\mu }{q}-\frac{\lambda }{p}+\frac{n}{q}-\frac{n}{p}+1>0$

2) For, $1\le q<p<\infty$

$\lambda <p-1;\frac{n}{q}-\frac{n}{p}+1\ge 0;\frac{\mu }{q}-\frac{\lambda }{p}+\frac{1}{q}-\frac{1}{p}+1>0$

3) For, $q>1$

$\mu \left(q^{\prime }-1\right)<1$, by the simple modifications of the example in [11]. Moreover, the monotonicity condition are satisfied:

$\begin{array}{l}{\displaystyle \underset{ij}{\sum }}\left({\sigma }_{ij}\left(x\mathrm{,}\eta \mathrm{,}{\xi }_I\right)-{\sigma }_{ij}\left(x\mathrm{,}\eta \mathrm{,}{{\xi }^{\prime }}_I\right)\right)\left({\xi }_{ij}-{{\xi }^{\prime }}_{ij}\right)\\ =\omega \left(x\right){\displaystyle \underset{ij}{\sum }}\left({\left|{\xi }_{ij}\right|}^{p-1}sng\left({\xi }_{ij}\right)-{\left|{{\xi }^{\prime }}_{ij}\right|}^{p-1}sng\left({{\xi }^{\prime }}_{ij}\right)\right)\left({\xi }_{ij}-{{\xi }^{\prime }}_{ij}\right)\ge 0\end{array}$

for almost all $x\in \Omega$ and for all, $\xi \mathrm{,}{\xi }^{\prime }\in M^n$. This last inequality cannot be strict, since for ${\xi }_{I^c}\ne {{\xi }^{\prime }}_{I^c}$ with ${\xi }_{i_{0}j}\ne {{\xi }^{\prime }}_{i_{0}j}$ for all $j=1,2,\cdots ,m$. But ${\xi }_{ij}={{\xi }^{\prime }}_{ij}$ for $i\in I^c$, $i\ne i_0$, $j=1,2,\cdots ,m$ the corresponding expression is Zero.