Existence and uniqueness of entropy solution of a nonlinear elliptic equation in anisotropic Sobolev–Orlicz space

Abstract

Our objective in this paper is to study a certain class of anisotropic elliptic equations with the second term, which is a low-order term and non-polynomial growth; described by an N-uplet of N-function satisfying the $\Delta _{2}$-condition in the framework of anisotropic Orlicz spaces. We prove the existence and uniqueness of entropic solution for a source in the dual or in $L^{1}$, without assuming any condition on the behaviour of the solutions when x tends towards infinity. Moreover, we are giving an example of an anisotropic elliptic equation that verifies all our demonstrated results.

1 Introduction

In this paper, we focused on the study of existence and uniqueness solution to anisotropic elliptic non-linear equation, driven by low-order term and non-polynomial growth; described by n-uplet of N-function satisfying the $\Delta _{2}$-condition, in Sobolev–Orlicz anisotropic space $\mathring{W}_{B}^{1} ( \Omega ) = \overline{C^{\infty } ( \Omega )}^{\mathring{W}_{B}^{1} ( \Omega )}$. To be more precise, $\Omega$ is an unbounded domain of $\mathbb {R}^{N}$, $N \ge 2$, we study the following equation:

$$\begin{aligned} ( \mathcal {P} ) {\left\{ \begin{array}{ll} \, A( u ) + \displaystyle \sum _{i = 1}^{N} b_{i} ( x, u, \nabla u ) = f( x ) &\quad \text {in} \,\,\Omega , \\ \, u = 0 &\quad \text {on} \, \partial \Omega . \end{array}\right. } \end{aligned}$$

where $A( u ) = \displaystyle \sum\nolimits_{i = 1}^{N} ( \, a_{i} ( x, u, \nabla u )\,)_{x_{i}}$ is a Leray–Lions operator defined from $\mathring{W}_{B}^{1} ( \Omega )$ into its dual, $B( \theta ) = ( \, B_{1} ( \theta ), \ldots , B_{N} ( \theta ) \, )$ are N-uplet Orlicz functions that satisfy the $\Delta _{2}$-condition, and for $i = 1, \ldots , N, \, b_{i} ( x, u, \nabla u ) : \Omega \times \mathbb {R} \times \mathbb {R}^{N} \longrightarrow \mathbb {R}$ the Carathéodory functions that do not satisfy any sign condition and the growth described by the vector N-function $B( \theta )$. In the recent studies, specifically the case of bounded domain $\Omega$ which is a well known for operators with polynomial, non-standard and non-polynomial growth (described by N-function). We refer the reader to [13,14,15,16,17,18, 28, 33] for the classical case, and for the Sobolev-Spaces with variable exponents Mihăilescu, M. et al. in [35]; were they proved the existence of solutions on the following nonhomogeneous anisotropic eigenvalue problem:

$$\begin{aligned} ( \mathcal {P} ) {\left\{ \begin{array}{ll} \, \displaystyle \sum _{i = 1}^{N} \partial _{x_{i}} ( |\, \partial _{x_{i}} u \,|^{p_{i} ( x ) - 2} \partial _{x_{i}} u ) = \lambda |\, u\,|^{q( x ) - 2} u &\quad \text {in} \,\,\Omega , \\ \, u = 0 &\quad \text {on} \, \partial \Omega . \end{array}\right. } \end{aligned}$$

where $\Omega \subset \mathbb {R}^{N} \, ( N \ge 3 )$ is a bounded domain with smooth boundary, $\lambda$ is a positive number and $p_{i}, \, q$ are continuous functions on $\bar{\Omega }$ such as $2 \le p_{i} ( x ) < N$ and $q( x ) > 1$ for any $x \in \bar{\Omega }$ and $i = \{ \, 1, \ldots , N \, \}$. For more detail we refer the reader to [36, 37], and [2, 3, 5, 9, 10, 25,26,27, 32, 34, 38, 39] for Orlicz Spaces.

In the case where $\, \Omega \,$ is an unbounded domain, without any assumption on the behaviour of solution where $\, | x | \, \longrightarrow \, + \infty \, .$ The existing result has been established by Brézis [19] for the semi-linear equation:

$$\begin{aligned} - \Delta u \, + \, | u |^{p_{0} \, - \, 2} \, u \, = \, f( x ). \end{aligned}$$

$\text{ Where } \,\,x \,\in \mathbb {R}^{N}, \,\, p_{0} \, > \, 2, \,\, f\, \in L_{1, loc} ( \mathbb {R}^{N} )$. Karlson and Bendahmane in [8] solved the problem $\mathcal {( P )} \,$ in the classic case such as $b ( x, u, \nabla u ) \, = \, \text{ div } ( g( u ) ),$ with g(u) has a growth like $| \, u \, |^{q - 1} , \, q \in ( 1, p_{0} \, - \, 1 ).$ For more result we refer to [24]. In the Sobolev-Spaces with variable exponent, in [20] have demonstrated the existence of solutions to the following problem: $\Delta _{p( x )} u + |\, u\,|^{p( x ) - 2} u = f( x, u ) \text{ in } \,\, \Omega = \mathbb {R}^{N}$, in both situations were $p: \Omega \longrightarrow \mathbb {R}$ is a log-Hölder continuous functions satisfying

$$\begin{aligned} 1<p^{-} = \inf _{x \in \Omega } p( x ) \le p^{+} = \sup _{x \in \Omega } p( x ) < \min \, \{\, n, \, \frac{n p^{-}}{n - p} \} \end{aligned}$$

and $f( x, u ) = \lambda f_{1}( x, u ) \,-\, \delta f_{2} ( x, u ) \,+\, \eta f_{3} ( x, u )$ with $\lambda , \delta , \eta$ as real positive parameters, $f_{1}, f_{2}, f_{3} : \Omega \times \mathbb {R} \rightarrow \mathbb {R}$ are Carathéodory functions with subcritical growth. The dependence among the parameters makes $f_{1}$ a perturbation of $f_{3}$ and, in turn, $f_{2}$ a perturbation of $f_{1}$. For more result we refer to the work of Aharrouch Benali and al. [6], for the Orlicz-Anisotropic Spaces L. M. Kozhevnikova [30] solved the problem $\mathcal {( P )} \,$ without the lower order $b_{i} ( x, u, \nabla u ) \,$ and $\, f( x ) \, = \, 0,$ we also cite [7, 23, 29, 31] for more detail.

Our goal, in this paper, is to show the existence and uniqueness of entropy solution for the equations $( \mathcal {P} )$; governed with growth and described by an N-uplet of N-functions satisfying the $\Delta _{2}$-condition. The function $b_{i} ( x, u, \nabla u ) \,$ does not satisfy any sign condition and the source f is merely integrable, within the fulfilling of anisotropic Orlicz spaces. An approximation procedure and some a priori estimates are used to solve the problem, the challenges that we had were due to behaviour of solution near infinity.

Definition 1.1

A measurable function $u: \, \Omega \longrightarrow \mathbb {R}$ is called an entropy solution of the problem $( \mathcal {P} )$ if it satisfies the following conditions: $1/ \quad u \in \mathcal {T}_{0}^{1, B} ( \Omega ) = \{\, u : \Omega \longrightarrow \mathbb {R} \,\, \text{ measurable }, \, T_{k} ( u ) \in \mathring{W}_{B}^{1} ( \Omega ) \,\, \text{ for } \text{ any } \, k > 0\}$ $2/ \quad b( x, u, \nabla u ) \in L^{1} ( \Omega )$ $3/ \quad \text{ For } \text{ any } \,\, k > 0$

$$\begin{aligned}&\int _{\Omega } a( x, u, \nabla u ) \cdot \nabla T_{k} ( u - \xi ) \,\, dx + \int _{\Omega } b( x, u, \nabla u ) \cdot T_{k} ( u - \xi ) \,\, dx\\&\quad \le \int _{\Omega } f( x ) \cdot T_{k} ( u - \xi ) \,\, dx \quad \forall \, \xi \in \mathring{W}_{B}^{1} ( \Omega ) \cap L^{\infty } ( \Omega ). \end{aligned}$$

The paper is organized as follows: in Sect. 2, we recall the most important and relevant properties and notation about N-functions and the space of Sobolev–Orlicz anisotropic, in Sect. 3, we show the existence of entropy solutions for the problem $(\mathcal {P} )$ in an unbounded domain, in Sect. 4, we demonstrate the uniqueness of the solution to the problem $( \mathcal {P} )$ in an unbounded domain and in Sect. 5 appendix.

2 Framework space: notations and basic properties

In this section, we briefly review some basic facts about Sobolev–Orlicz anisotropic space which we will need in our analysis of the problem $\mathcal {P}$. A comprehensive presentation of Sobolev–Orlicz anisotropic space can be found in the work of M.A Krasnoselskii and Ja. B. Rutickii [32] and [23].

Definition 2.1

We say that $B: \mathbb {R}^{+} \longrightarrow \mathbb {R}^{+}$ is a N-function if B is continuous, convex, with $B( \theta )> 0 \,\,\, \text{ for } \,\,\, \theta > 0$, $\frac{B( \theta )}{\theta } \rightarrow 0 \,\,\, \text{ when } \,\,\, \theta \rightarrow 0$ and $\frac{B( \theta )}{\theta } \rightarrow \infty \,\,\, \text{ when } \,\,\, \theta \rightarrow \infty .$ This N-function B admit the following representation: $B( \theta ) = \displaystyle \int\nolimits _{0}^{\theta } b( t ) \,\, dt$, with $b: \mathbb {R}^{+} \longrightarrow \mathbb {R}^{+}$ which is an increasing function on the right, with $b( 0 ) = 0$ in the case $\theta > 0$ and $b( \theta ) \longrightarrow \infty$ when $\theta \longrightarrow \infty$. Its conjugate is noted by $\bar{B} ( \theta ) = \displaystyle \int\nolimits _{0}^{|\, \theta \,|} q( t ) \,\, dt$ with q also satisfies all the properties already quoted from b, with

$$\begin{aligned} \bar{B} ( \theta ) = \sup _{\mu \ge 0} \,( \, \mu \, | \, \theta \, | - B( \mu ) \, ) ,\quad \theta > 0. \end{aligned}$$

(1)

The Young’s inequality is given as follow

$$\begin{aligned} \forall \theta , \, \mu > 0 \quad \theta \, \mu \le B( \mu ) + \bar{B} ( \theta ). \end{aligned}$$

(2)

Definition 2.2

The N-function $B( \theta )$ satisfies the $\Delta _{2}$-condition if $\, \exists c > 0, \, \theta _{0} \ge 0$ such as

$$\begin{aligned} B( 2 \, \theta ) \le c \, B( \theta ) \quad |\,\theta \,| \ge \theta _{0}. \end{aligned}$$

(3)

This definition is equivalent to, $\forall k> 1 , \,\,\, \exists \,\, c( k ) > 0$ such as

$$\begin{aligned} B ( K \, \theta ) \le c( K ) \, B( \theta ) \quad \text{ for } \quad |\,\theta \,| \ge \theta _{0}. \end{aligned}$$

(4)

Definition 2.3

The N-function $B( \theta )$ satisfies the $\Delta _{2}$-condition as long as there exists positive numbers $c \, > \, 1$ and $\theta _{0} \ge 0$ such as for $\theta \ge \theta _{0}$ we have

$$\begin{aligned} \theta \, b( \theta ) \le c \, B( \theta ). \end{aligned}$$

(5)

Also, each N-function $B( \theta )$ satisfies the inequality

$$\begin{aligned} B( \mu + \theta ) \le c \, B( \theta ) + c \, B( \mu ) \quad \theta , \,\, \mu \ge 0. \end{aligned}$$

(6)

We consider the Orlicz space $L_{B} ( \Omega )$ provided with the norm of Luxemburg given by

$$\begin{aligned} || \, u \, ||_{B, \, \Omega } = \inf \, \{ \, k > 0 \,/\,\, \int _{\Omega } B \bigg ( \, \frac{u( x )}{k} \, \bigg ) \,\, dx \le 1 \, \}. \end{aligned}$$

(7)

According to [32] we obtain the inequalities

$$\begin{aligned} \int _{\Omega } B \bigg ( \, \frac{u( x )}{||\, u\,||_{B, \, \Omega }} \, \bigg ) \,\, dx \le 1 \end{aligned}$$

(8)

and

$$\begin{aligned} ||\, u\,||_{B, \, \Omega } \le \int _{\Omega } B( u ) \,\, dx + 1. \end{aligned}$$

(9)

Moreover, the Hölder’s inequality holds and we have for all $u \in L_{B} ( \Omega )$ and $v \in L_{\bar{B}} ( \Omega )$

$$\begin{aligned} \big | \, \int _{\Omega } u( x ) \, v( x ) \,\, dx \, \big | \le 2 \, ||\, u\,||_{B, \Omega } \cdot ||\, v \,||_{\bar{B}, \, \Omega }. \end{aligned}$$

(10)

In [32] and [23], if $P( \theta )$ and $B( \theta )$ are two N-functions such as $P( \theta ) \ll B( \theta )$ and $\text{ meas } \, \Omega < \infty$, then $L_{B} ( \Omega ) \subset L_{P} ( \Omega )$, furthermore

$$\begin{aligned} ||\, u\,||_{P, \Omega } \le A_{0} \, ( \, \text{ meas } \, \Omega \,) \, ||\, u\,||_{B, \Omega } \quad u \in L_{B} ( \Omega ). \end{aligned}$$

(11)

And for all N-functions $B( \theta )$, if $\text{ meas } \, \Omega < \infty$, then $L_{\infty } ( \Omega ) \subset L_{B} ( \Omega )$ with

$$\begin{aligned} ||\, u\,||_{B, \Omega } \le A_{1} \, ( \, \text{ meas } \, \Omega \,) \, ||\, u\,||_{\infty , \Omega } \quad u \in L_{B} ( \Omega ). \end{aligned}$$

(12)

Also for all N-functions $B( \theta )$, if $\text{ meas } \, \Omega < \infty$, then $L_{B} ( \Omega ) \subset L^{1} ( \Omega )$ with

$$\begin{aligned} ||\, u\,||_{1, \Omega } \le A_{2} \, ||\, u\,||_{B, \Omega } \quad u \in L_{B} ( \Omega ). \end{aligned}$$

(13)

We define for all N-functions $B_{1} ( \theta ), \ldots , B_{N} ( \theta )$ the space of Sobolev–Orlicz anisotropic $\mathring{W}_{B}^{1} ( \Omega )$ as the adherence space $C_{0}^{\infty } ( \Omega )$ under the norm

$$\begin{aligned} ||\, u\,||_{\mathring{W}_{B}^{1} ( \Omega )} = \sum _{i = 1}^{N} \, ||\, u_{x_{i}} \,||_{B_{i}, \, \Omega }. \end{aligned}$$

(14)

Definition 2.4

A sequence $\{ \, u_{m}\,\}$ is said to converge modularly to u in $\mathring{W}_{B}^{1} ( \Omega )$ if for some $k > 0$ we have

$$\begin{aligned} \int _{\Omega } B \bigg ( \, \frac{u_{m} - u}{k} \, \bigg ) \,\, dx \longrightarrow 0 \quad \text{ as } \quad m \longrightarrow \infty . \end{aligned}$$

(15)

Remark 2.5

Since B satisfies the $\Delta _{2}$-condition, then the modular convergence coincide with the norm convergence.

Proposition 2.6

$$\begin{aligned} \theta \, B'( \theta ) \, = \, \bar{B} ( \, B' ( \theta ) \, ) \, + \, B( \theta ) \,, \theta > 0, \end{aligned}$$

(16)

with $\, B' \,$ is the right derivative of the N-function $\, B (\theta )$.

Proof

By (2), we take $\mu \, = \, B'( \theta )$, then we obtain

$$\begin{aligned} B'( \theta ) \, \theta \, \le B( \theta ) \, + \, \bar{B} ( B' ( \theta ) ) \end{aligned}$$

and by Ch. I [32], we get the result. $\square$

Let $\omega \subset \Omega$, be a bounded domain in $\mathbb {R}^{N}$. The following Lemmas are true:

Lemma 2.7

[27] For all $u \in \mathring{W}^{1}_{L_{B}} ( \omega )$ with $\text{ meas } \, \omega < \infty$, we have

$$\begin{aligned} \int _{\omega } B \bigg ( \, \frac{|\,u\,|}{\lambda } \, \bigg ) \,\, dx \le \int _{\omega } B( \, |\, \nabla u \,|\, ) \,\, dx \end{aligned}$$

where $\lambda = \text{ diam } ( \omega )$, is the diameter of $\omega$.

Note by $\, h( t ) \, = \, \bigg (\displaystyle \prod _{i = 1}^{N} \frac{B_{i}^{-1} ( t ) }{t} \, \bigg )^{\frac{1}{N}} \,$ and we assume that $\displaystyle \int\nolimits _{0}^{1} \frac{h( t )}{t} \,\, dt \,$ converge, so we consider the N-functions $\, B^{*} ( z ) \,$ defined by $\, ( B^{*} )^{-1} ( z ) \, = \,\displaystyle \int\nolimits _{0}^{| \, z \, |} \frac{h( t )}{t} \,\, dt \, .$

Lemma 2.8

[29] Let $\, u \, \in \mathring{W}_{B}^{1} ( \omega )$. If

$$\begin{aligned} \int _{1}^{\infty } \frac{h( t ) }{t} \,\, dt \, = \, \infty , \end{aligned}$$

(17)

then, $\, \mathring{W}_{B}^{1} ( \omega ) \subset L_{B^{*}} ( \omega )$ and $||\, u\,||_{B^{*}, \omega } \le \frac{N - 1}{N} \, ||\, u\,||_{\mathring{W}_{B}^{1} ( \omega )}.$ If

$$\begin{aligned} \int _{1}^{\infty } \frac{h( t ) }{t} \,\, dt \, \le \, \infty , \end{aligned}$$

then, $\, \mathring{W}_{B}^{1} ( \omega ) \subset L_{\infty } ( \omega )$ and $||\, u\,||_{\infty , \omega } \le \beta \, ||\, u\,||_{\mathring{W}_{B}^{1} ( \omega )},$ with $\beta = \displaystyle \int\nolimits _{0}^{\infty } \frac{h( t ) }{t} \,\, dt.$

Lemma 2.9

Suppose that conditions (20)–(23) are satisfied, and let $( u^{m} )_{m \in \mathbb {N}}$ be sequence in $\mathring{W}_{B}^{1} ( \omega )$ such as

• (a) $u^{m} \rightharpoonup u \,\, \text{ in } \,\, \mathring{W}_{B}^{1} ( \omega ).$

• (b) $a^{m} ( x, u^{m}, \nabla u^{m} ) \,\, \text{ is } \text{ bounded } \text{ in } \,\, L_{\bar{B}} ( \omega ).$

• (c) $\displaystyle \sum \nolimits _{i = 1}^{N} \int\nolimits _{\omega } \bigg [ \, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x, u^{m}, \nabla u \chi _{s} ) \, \bigg ] \cdot ( \nabla u^{m} - \nabla u \, \chi _{s} ) \,\,dx \longrightarrow 0 \text{ as } \,\, m \rightarrow + \infty , \,\, s \rightarrow \infty .$ Where $\chi _{s}$ is the characteristic function of $\omega ^{s} = \{ \, x \in \omega : \, |\, \nabla u\,| \le s \, \}.$ Then,

  $$\begin{aligned} \nabla u^{m} \longrightarrow \nabla u \,\, \text{ a.e } \text{ in } \,\, \omega , \end{aligned}$$

  (18)

  and

  $$\begin{aligned} B( |\, \nabla u^{m}\,| ) \longrightarrow B( |\, \nabla u\,| ) \,\, \text{ in } \,\, L^{1} ( \omega ). \end{aligned}$$

  (19)

Proof

Let $\vartheta > 0$ fixed and $s > \vartheta$, then from (21) we have

$$\begin{aligned} 0&\le \sum _{i = 1}^{N} \int _{\omega ^{\vartheta }} \bigg [ \, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x, u^{m}, \nabla u ) \,\bigg ] \cdot ( \nabla u^{m} - \nabla u ) dx \\&= \sum _{i = 1}^{N} \int _{\omega ^{s}} \bigg [ \, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x, u^{m}, \nabla u \, \chi _{s} ) \,\bigg ] \cdot ( \nabla u^{m} - \nabla u\, \chi _{s} ) dx \\&\le \sum _{i = 1}^{N} \int _{\omega } \bigg [ \, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x, u^{m}, \nabla u \, \chi _{s} ) \,\bigg ] \cdot ( \nabla u^{m} - \nabla u\, \chi _{s} ) dx. \end{aligned}$$

According to (c), we get

$$\begin{aligned} \lim _{m \rightarrow \infty } \sum _{i = 1}^{N} \int _{\omega ^{\vartheta }} \bigg [ \, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x, u^{m}, \nabla u ) \,\bigg ] \cdot ( \nabla u^{m} - \nabla u ) \,\, dx = 0. \end{aligned}$$

Proceeding as in [4], we obtain

$$\begin{aligned} \nabla u^{m} \longrightarrow \nabla u \,\, \text{ a.e } \text{ in } \,\, \omega . \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m}) \cdot \nabla u^{m} dx &=\sum _{i = 1}^{N} \int _{\omega } \bigg [ \, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x, u^{m}, \nabla u \, \chi _{s} ) \,\bigg ] \\&\quad \times \, (\nabla u^{m} - \nabla u\, \chi _{s} ) \,\, dx\\&\quad +\, \sum _{i = 1}^{N} \int _{\omega } a_{i}^{m} ( x, u^{m}, \nabla u \, \chi _{s} ) \cdot (\nabla u^{m} - \nabla u \, \chi _{s} ) \cdot dx\\&\quad +\, \sum _{i = 1}^{N} \int _{\omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u \chi _{s} dx, \end{aligned}$$

using (b) and (18), we obtain

$$\begin{aligned} \sum _{i = 1}^{N} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \rightharpoonup \sum _{i = 1}^{N} a_{i} ( x, u, \nabla u ) \,\, \text{ weakly } \text{ in } \,\, ( \, L_{\bar{B}} ( \omega ) \, )^{N}. \end{aligned}$$

Therefore

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} )\, \nabla u \, \chi _{s} \,\, dx \longrightarrow \sum _{i = 1}^{N} \int _{\omega } a_{i} ( x, u, \nabla u ) \cdot \nabla u \end{aligned}$$

as $m \rightarrow \infty , \,\, s \rightarrow \infty$. So,

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\omega } \bigg [ \, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x, u^{m}, \nabla u \, \chi _{s} ) \,\bigg ] \cdot ( \nabla u^{m} - \nabla u\, \chi _{s} ) \,\, dx \longrightarrow 0, \end{aligned}$$

and

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\omega } a_{i}^{m} ( x, u^{m}, \nabla u \, \chi _{s} ) \cdot ( \nabla u^{m} - \nabla u \, \chi _{s} ) \cdot dx \longrightarrow 0. \end{aligned}$$

Thus,

$$\begin{aligned} \lim _{m \rightarrow \infty } \sum _{i = 1}^{N} \int _{\omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \,\, dx = \sum _{i = 1}^{N} \int _{\omega } a_{i} ( x, u, \nabla u ) \cdot \nabla u \,\, dx, \end{aligned}$$

from (22) and vitali’s Theorem, we get

$$\begin{aligned} \bar{a} \, \sum _{i = 1}^{N} \int _{\omega } B_{i} ( |\, \nabla u^{m} \,| ) \,\, dx - \int _{\omega } \phi ( x ) \,\, dx \ge \bar{a} \, \sum _{i = 1}^{N} \int _{\omega } B_{i} ( |\, \nabla u \,| )\,\, dx - \int _{\omega } \phi ( x ) \,\, dx. \end{aligned}$$

Consequently, by Lemma 2.6 in [27], we get

$$\begin{aligned} B( |\, \nabla u^{m} \, | ) \longrightarrow B( |\, \nabla u \, | ) \,\, \text{ in } \,\, \mathring{W}_{B}^{1} ( \omega ). \end{aligned}$$

Thanks to Lemma 1 in [29], we have

$$\begin{aligned} B( |\, \nabla u^{m} \,| ) \longrightarrow B( |\, \nabla u \,| ) \,\, \text{ in } \,\, L^{1} ( \omega ). \end{aligned}$$

$\square$

3 Existence result in unbounded domain

In this section, we assume they have non-negative measurable functions $\phi , \, \varphi \in L^{1} ( \Omega )$ and $\bar{a}, \, \tilde{a}$ are two positive constants such that

$$\begin{aligned}&\sum _{i = 1}^{N} |\, a_{i} ( x, s, \xi ) \, | \le \tilde{a} \, \sum _{i = 1}^{N} \bar{B}_{i}^{-1} B_{i} ( |\, \xi \,| ) + \varphi ( x ), \end{aligned}$$

(20)

$$\begin{aligned}&\sum _{i = 1}^{N} \big ( \, a_{i} ( x, s, \xi ) - a_{i} ( x, s, \xi ^{'} ) \, \big ) \cdot ( \xi _{i} - \xi _{i}^{'} ) > 0, \end{aligned}$$

(21)

$$\begin{aligned}&\sum _{i = 1}^{N} a_{i} ( x, s, \xi ) \cdot \xi _{i} > \bar{a} \, \sum _{i = 1}^{N} B_{i} ( | \, \xi \,| ) - \phi ( x ), \end{aligned}$$

(22)

and there exists $h \in L^{1} ( \Omega )$ and $l : \mathbb {R} \longrightarrow \mathbb {R}^{+}$ a positive continuous functions such that $l \in L^{1} ( \mathbb {R} ) \cap L^{\infty } ( \mathbb {R} ).$

$$\begin{aligned} \sum _{i = 1}^{N} | \, b_{i} ( x, s, \xi ) \, | \le l( s ) \cdot \sum _{i = 1}^{N} B_{i} ( |\, \xi \, | ) + h( x ). \end{aligned}$$

(23)

Theorem 3.1

Let $\Omega$ be an unbounded domain of $\mathbb {R}^{N}$. Under assumptions (20)–(23), there exists a least one entropy solution of the problem $( \mathcal {P} )$ on the sense of Definition 1.1.

Proof

Let $\Omega ( m ) = \{ \, x \in \Omega : \, | \, x \,| \le m \, \}$ and $f^{m} ( x ) = \frac{f( x )}{1 + \frac{1}{m} \, | \, f( x ) \, |} \cdot \chi _{ \Omega ( m )} .$

We have $f^{m} \longrightarrow f \,\, \text{ in } \,\, L^{1} ( \Omega ),\, m \rightarrow \infty , \, |\, f^{m} ( x )\,| \le |\, f( x )\,|$ and $| f^{m} | \le m \chi _{\Omega ( m )}.$

$$\begin{aligned} a^{m} ( x, s, \xi ) = ( \, a^{m}_{1} ( x, s, \xi ), \ldots , a_{N}^{m} ( x, s, \xi ) \, ) \end{aligned}$$

where $a_{i}^{m} ( x, s, \xi ) = a_{i} ( x, T_{m} ( s ), \xi ) \,\, \text{ for } \,\, i = 1, \ldots , N.$

$$\begin{aligned} b^{m} ( x, s, \xi ) = T_{m} ( \, b( x, s, \xi ) \, ) \cdot \chi _{\Omega ( m )} \end{aligned}$$

and for any $v \in \mathring{W}_{B}^{1} ( \Omega )$, we consider the following approximate equations

$$\begin{aligned} ( \mathcal {P}_{m} ) : \int _{\Omega } a( x, T_{m}( u^{m} ), \nabla u^{m} ) \, \nabla v \, dx + \int _{\Omega } b^{m} ( x, u^{m}, \nabla u^{m} ) \, v \, dx = \int _{\Omega } f^{m} \, v \, dx. \end{aligned}$$

For the proof. See Appendix 5. We divide our proof in six steps.

Step 1 A priori estimate of $\{ \, u^{m}\,\}$.

Proposition 3.2

Suppose that the assumptions (20)–(23) hold true, and let $( u^{m} )_{m}$ be a solution of the approximate problem $( \mathcal {P}_{m} )$. Then, for all $k > 0$, there exists a constant $c\cdot k$ ( not depending on m ), such that

$$\begin{aligned} \int _{\Omega } B( |\, \nabla T_{k} ( u^{m} ) \,| ) \le c \cdot k \end{aligned}$$

Proof

Taking $v \, = \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} )$, as a test function with $\, G( s ) = \displaystyle \int\nolimits _{0}^{s} \frac{l( t ) }{\bar{a}} \,\, dt \,$ and $\, \bar{a} \,$ is the coercivity constant, we obtain

$$\begin{aligned}&\sum _{i = 1}^{N} \, \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla ( \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) ) \,\, dx \\&\qquad + \, \sum _{i = 1}^{N} \, \int _{\Omega } b_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) \,\, dx \\&\quad \le \,\int _{\Omega } f^{m} \cdot \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) \,\, dx. \end{aligned}$$

Then,

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \exp ( G( u^{m} ) ) \, \nabla T_{k} ( u^{m} ) ) \,\, dx\\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} )\cdot \nabla u^{m}\cdot \frac{l( u^{m} )}{\bar{a}} .\exp ( G( u^{m} ) )T_{k} ( u^{m} )dx \\&\quad \le \, \sum _{i = 1}^{N} \, \int _{\Omega } | \, b_{i}^{m} ( x, u^{m}, \nabla u^{m} )\, | \cdot \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) \,\, dx + \, \int _{\Omega } f^{m} \cdot \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) \,\, dx\\&\quad \le \sum _{i = 1}^{N} \int _{\Omega } \big [ h( x ) \, + \, l( u^{m} ) \cdot B_{i} ( \nabla u^{m} ) \, \big ] \cdot \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) \,\, dx \\&\qquad +\, \int _{\Omega } f^{m} \cdot \exp ( G( u^{m} ) ) \times \, T_{k} (u^{m} ) \,\, dx \\&\quad \le \, \sum _{i = 1}^{N} \int _{\Omega } l( u^{m} ) \cdot B_{i} ( \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) \,\, dx \\&\qquad + \,\int _{\Omega } \big ( \, f^{m} \, + \, h( x ) \, \big )\cdot \exp ( G( u^{m} ) ) \cdot T_{k} ( u^{m} ) \,\, dx , \end{aligned}$$

so,

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\{ \, \Omega : | u^{m} | \, < \, k \, \}} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \exp ( G( u^{m} ) ) \,\, dx \\&\quad \le \, \int _{\Omega } \big [ \, f^{m} ( x ) \, + \, h( x ) \, + \, \phi ( x ) \, \frac{l( u^{m} )}{\bar{a}} \, \big ] \cdot \exp ( G( u^{m} ) ) \, T_{k} ( u^{m} ) \,\,dx \end{aligned}$$

by (22), we get

$$\begin{aligned}&\bar{a} \, \sum _{i = 1}^{N} \int _{\{ \Omega \, : \, | \, u^{m} \, | \, \le \, k \}} B_{i} ( \nabla u^{m} ) \, \exp ( G( u^{m} ) ) \,\, dx \\&\quad \le \, \int _{\{ \Omega \, : \, | \, u^{m} \, | \, \le \, k \}} \phi ( x ) \, \exp ( G( u^{m} ) ) \,\, dx \\&\qquad +\, \, \int _{\Omega } \big [ \, f^{m} ( x ) \, + \, h( x ) \, + \, \phi ( x ) \, \frac{l( u_{m} )}{\bar{a}} \, \big ] \cdot \exp ( G( u^{m} ) ) \, T_{k} ( u^{m} ) \,\,dx, \end{aligned}$$

since $\, \phi , \, h\, \text{ and }\, f^{m} \, \in \, L^{1} ( \Omega ) \,$, and the fact that $\exp ( G( \pm \infty ) ) \, \le \, \exp \bigg ( \, \frac{|| \, l \, ||_{L^{1} ( \Omega )}}{\bar{a}} \, \bigg ),$ we deduce that,

$$\begin{aligned} \int _{\{\, \Omega : \, | u^{m} | \, < \, k \, \}} B( \nabla T_{k} ( u^{m} ) ) \,\, dx \, \le \, k \cdot c \,\,\,\,\,\, k\, > \, 0 . \end{aligned}$$

Finally

$$\begin{aligned} \int _{\Omega } B( \nabla T_{k} ( u^{m} ) ) \,\, dx \, \le \, k\cdot c\,\,\,\,\,\, k\, > \, 0 . \end{aligned}$$

$\square$

Step 2 Almost everywhere convergence of $\{ \, u^{m}\,\}$.

Lemma 3.3

For all $u^{m}$ measurable function on $\Omega$, we have

$$\begin{aligned} \text{ meas } \{ \, x \in \Omega , \, |\, u^{m} \,| > k \, \} \longrightarrow 0. \end{aligned}$$

Proof

According to Lemma 2.7 and Lemma 2.8, we have

$$\begin{aligned} ||\, T_{k} ( u^{m} ) \,||_{B^{*}}&\le A \cdot ||\, \nabla T_{k} ( u^{m} ) \,||_{B} \nonumber \\&\le A \cdot \epsilon ( k ) \,\int _{\omega } B ( \, \nabla T_{k} ( u^{m} ) \,\, dx\nonumber \\&\le c\cdot k \cdot \epsilon ( k ) \quad \text{ for } \,\, k >1 \end{aligned}$$

(24)

with $\epsilon ( k ) \longrightarrow 0 \,\, \text{ as } \,\, k \longrightarrow \infty$.

Form (24) we have

$$\begin{aligned} B^{*} \bigg ( \, \frac{k}{||\, T_{k} ( u^{m} ) \,||_{B^{*}}} \, \bigg ) \, \text{ meas } \{ \, x \in \Omega : \, |\, u^{m} \,| \ge k \, \}&\le \int _{\Omega } B^{*} \bigg ( \, \frac{T_{k} ( u^{m} )}{||\, T_{k} ( u^{m} ) \,||_{B^{*}}} \, \bigg ) \,\, dx \\&\le \int _{\Omega } B^{*} \bigg ( \, \frac{k}{||\, T_{k} ( u^{m} ) \,||_{B^{*}}} \, \bigg ) \,\, dx \end{aligned}$$

by (24) again, we obtain

$$\begin{aligned} B^{*} \bigg ( \, \frac{k}{||\, T_{k} ( u^{m} ) \,||_{B^{*}}} \, \bigg ) \longrightarrow \infty \,\, \text{ as } \,\, k \longrightarrow \infty . \end{aligned}$$

Hence,

$$\begin{aligned} \text{ meas } \{ \,x \in \Omega : | \, u^{m}\,| \ge k \, \} \longrightarrow 0 \,\, \text{ as } \,\, k \longrightarrow \infty \,\,\, \text{ for } \text{ all } \,\,\, m \in \mathbb {N}. \end{aligned}$$

$\square$

Lemma 3.4

For all $u^{m}$ measurable function on $\Omega$, such that

$$\begin{aligned} T_{k} ( u^{m} ) \in \mathring{W}_{B}^{1} ( \Omega ) \quad \forall k \ge 1. \end{aligned}$$

We have,

$$\begin{aligned} \text{ meas } \{ \, \Omega : \, B( \nabla u^{m} ) \ge r \, \} \longrightarrow 0 \,\, \text{ as } \,\, r \longrightarrow \infty . \end{aligned}$$

Proof

$$\begin{aligned} \text{ meas } \{\, x \in \Omega : \, B( \nabla u^{m} ) \ge 0 \, \}&= \text{ meas } \{\,\{\, x \in \Omega : \, |\, u^{m}\,| \ge k\,\,\, B(\nabla u^{m} ) \ge r \, \} \\&\quad \cup \{ \, x \in \Omega : \, |\, u^{m}\,| < k \,\,\, B( \nabla u^{m} ) \ge r \,\} \,\} \end{aligned}$$

if we denote

$$\begin{aligned} g( r, k ) = \text{ meas } \{ \, x \in \Omega : \, |\, u^{m} \,| \ge k, \,\, B( \nabla u^{m} ) \ge r \, \} \end{aligned}$$

we have

$$\begin{aligned} \text{ meas } \{ \, x \in \Omega : \, |\, u^{m}\,| < k \,\,\, B( \nabla u^{m} ) \ge r \,\} = g( r, 0 ) - g( r, k ). \end{aligned}$$

Then,

$$\begin{aligned} \int _{\{\, x \,\in \,\Omega : \,|\, u^{m}\,| < k \, \}} B( \nabla u^{m} ) \,\, dx = \int _{0}^{\infty } \big ( \, g( r, 0 ) - g( r, k ) \, \big ) \,\, dr \le c\cdot k \end{aligned}$$

(25)

with $r \longrightarrow g( r, k )$ is a decreasing map. Then,

$$\begin{aligned} g( r, 0 )&\le \frac{1}{r} \, \int _{0}^{r} g( r, 0 ) \,\, dr \nonumber \\&\le \frac{1}{r} \, \int _{0}^{r} \big ( \, g( r, 0 ) - g( r, k ) \, \big )\,\, dr + \frac{1}{r} \, \int _{0}^{r} g( r, k ) \,\, dr \nonumber \\&\le \frac{1}{r} \, \int _{0}^{r} \big ( \, g( r, 0 ) - g( r, k ) \, \big )\,\, dr + g( 0, k ) \end{aligned}$$

(26)

combining (25) and (26), we obtain

$$\begin{aligned} g( r, 0 ) \le \frac{c\cdot k}{r} + g( 0, k ) \end{aligned}$$

by Lemma 2.7,

$$\begin{aligned} \lim _{k \rightarrow \infty } g( 0, k ) = 0. \end{aligned}$$

Thus

$$\begin{aligned} g( r, 0 ) \longrightarrow 0 \,\, \text{ as } \,\, r \longrightarrow \infty . \end{aligned}$$

$\square$

We have now to prove the almost everywhere convergence of $\{\, u^{m} \,\}$

$$\begin{aligned} u^{m} \longrightarrow u \,\, \text{ a.e } \text{ in } \,\, \Omega . \end{aligned}$$

(27)

Let $g( k ) = \displaystyle \sup _{m \in \mathbb {N}} \, \text{ meas } \{\, x \in \Omega : \, | \, u^{m} \,| > k \,\} \longrightarrow 0 \,\, \text{ as } \,\, k \longrightarrow \infty .$

Since $\Omega$ is unbounded domain in $\mathbb {R}^{N}$, we define $\eta _{R}$ as

$$\begin{aligned} \eta _{R} ( r ) = {\left\{ \begin{array}{ll} 1 &\quad \text {if }\, r< R , \\ R + 1 - r &\quad \text {if }\, R \le r < R + 1,\\ 0 &\quad \text {if}\, r \ge R + 1 . \end{array}\right. } \end{aligned}$$

For $R, \, k > 0,$ we have by (6)

$$\begin{aligned} \int _{\Omega } B( \nabla \eta _{R} ( |\, x\,| ) \cdot T_{k} ( u^{m} ) ) \,\, dx&\le \,c \, \int _{\{\, x\, \in \, \Omega : |\,u^{m}\,| < k \,\}} B( \nabla u^{m} )\,\, dx \\&\quad +\, c \, \int _{\Omega } B( T_{k} ( u^{m} ) \cdot \nabla \eta _{R} ( |\, x\,| ) \,\,dx \\&\le c( k, \, R ), \end{aligned}$$

which implies that the sequence $\{\, \eta _{R} ( |\, x \,| ) \, T_{K} ( u^{m} ) \, \}$ is bounded in $\mathring{W}_{B}^{1} ( \Omega ( R + 1 ) )$ and by embedding Theorem, for $P \ll B$ we have

$$\begin{aligned} \mathring{W}_{B}^{1} ( \Omega ( R + 1 ) ) \hookrightarrow L_{P} ( \Omega ( R + 1 ) ), \end{aligned}$$

and since $\eta _{R} = 1$ in $\Omega ( R ),$ we have

$$\begin{aligned} \eta _{R} \, T_{k} ( u^{m} ) \longrightarrow v_{k} \,\, \text{ in } \,\, L_{P} ( \Omega ( R + 1 ) ) \,\, \text{ as } \,\, m \longrightarrow \infty . \end{aligned}$$

For $k = 1, \ldots ,$

$$\begin{aligned} T_{k} ( u^{m} ) \longrightarrow v_{k} \,\, \text{ in } \,\, L_{P} ( \Omega ( R + 1 ) ) \,\, \text{ as } \,\, m \longrightarrow \infty , \end{aligned}$$

by diagonal process, we prove that there is $u : \Omega \longrightarrow \mathbb {R}$ measurable such that $u^{m} \longrightarrow u$ a.e in $\Omega$. This implies the (27).

Lemma 3.5

Let an N-functions $\bar{B} ( t )$ satisfy the $\Delta _{2}$-condition and $u^{m}, \,\, m = 1, \ldots , \infty ,$ and u be two functions of $L_{B} ( \Omega )$ such as

$$\begin{aligned} || \, u^{m} \, ||_{B} \le c \quad m = 1, 2, \ldots .\\ u^{m} \longrightarrow u \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega , \,\, m \longrightarrow \infty . \end{aligned}$$

Then,

$$\begin{aligned} u^{m} \rightharpoonup u \,\, \text{ weakly } \text{ in } \,\, L_{B} ( \Omega ) \,\, \text{ as } \,\, m \rightarrow \infty . \end{aligned}$$

Proof

See Lemma 1.3 in [34]. $\square$

Step 3 Weak convergence of the gradient.

Since $\mathring{W}_{B}^{1} ( \Omega )$ reflexive, then, there exists a subsequence

$$\begin{aligned} T_{k} ( u^{m} ) \rightharpoonup v \,\, \text{ weakly } \text{ in } \,\, \mathring{W}_{B}^{1} ( \Omega ), \,\, m \rightarrow \infty . \end{aligned}$$

And since,

$$\begin{aligned} \mathring{W}_{B}^{1} ( \Omega ) \hookrightarrow L_{B} ( \Omega ), \end{aligned}$$

we have

$$\begin{aligned} \nabla T_{k} ( u^{m} ) \rightharpoonup \nabla v \,\, \text{ in } \,\, L_{B} ( \Omega ) \,\, \text{ as } \,\, m \rightarrow \infty , \end{aligned}$$

since

$$\begin{aligned} u^{m} \longrightarrow u \,\, \text{ a.e } \text{ in } \,\, \Omega \,\, \text{ as } \,\, m \rightarrow \infty , \end{aligned}$$

we get

$$\begin{aligned} \nabla u^{m} \longrightarrow \nabla u \,\, \text{ a.e } \text{ in } \,\, \Omega \,\, \text{ as } \,\, m \rightarrow \infty . \end{aligned}$$

Then, we obtain for any fixed $k > 0$

$$\begin{aligned} \nabla T_{k} ( u^{m} ) \longrightarrow \nabla T_{k} ( u ) \,\, \text{ a.e } \text{ in } \,\, \Omega . \end{aligned}$$

Applying Lemma 3.5, we have the following weak convergence

$$\begin{aligned} \nabla T_{k} ( u^{m} ) \rightharpoonup \nabla T_{k} ( u ) \,\, \text{ in } \,\, L_{B} ( \Omega ) \,\, \text{ as } \,\, m \rightarrow \infty , \end{aligned}$$

for more detail see page 11 in [10].

Step 4 Strong convergence of the gradient.

For $j> k > 0$, we introduce the following function defined as

$$\begin{aligned} h_{j} (s) = {\left\{ \begin{array}{ll} 1 &\quad \text {if} \,| s| \le j , \\ 1 - |\, s - j \,| &\quad \text {if }\, j \le |s| \le j + 1 ,\\ 0 &\quad \text {if}\, s \ge j + 1. \end{array}\right. } \end{aligned}$$

and we show that the following assertions are true:

Assertion 1

$$\begin{aligned} \lim _{j \rightarrow \infty } \, \lim _{m \rightarrow \infty } \sum _{i = 1}^{N} \int _{\{\, j \le |\, u^{m}\,| \le j + 1 \, \}} a^{m}_{i} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \eta _{R} ( |\,x\,| ) \,\, dx = 0. \end{aligned}$$

(28)

Assertion 2

$$\begin{aligned} \nabla u^{m} \longrightarrow \nabla u \,\, \text{ a.e } \text{ in } \,\, \Omega ( m ). \end{aligned}$$

(29)

Proof

We take $v = \exp ( G( u^{m} ) )\, T_{1, j} ( u^{m} ) \,\eta _{R} ( |\, x\,|) = \exp ( G( u^{m} ) ) \, T_{1} ( u^{m} - T_{j} ( u^{m} ) ) \, \eta _{R} ( |\,x \,|)$ as a test function in the problem $( \mathcal {P}_{m} )$, we obtain

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla \bigg ( \, \exp ( G( u^{m} ) ) \cdot T_{1} ( u^{m} - T_{j} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,|) \, \bigg ) \,\, dx \\&\quad \le \sum _{i = 1}^{N} \int _{\Omega } |\,b_{i}^{m} ( x, u^{m}, \nabla u^{m} )\,| \cdot \exp ( G( u^{m} ) ) \cdot T_{1} ( u^{m} - T_{j} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,|) \,\, dx \\&\qquad +\, \int _{\Omega } f^{m} ( x ) \cdot \exp ( G( u^{m} ) ) \cdot T_{1} ( u^{m} - T_{j} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,|) \,\, dx \end{aligned}$$

according to (22) and (23) we deduce that

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\{\, j< |\, u^{m} \,| < j +1 \,\}} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \exp ( G( u^{m} ) ) \cdot \eta _{R} ( |\, x\,| )\,\, dx \\&\quad \le \int _{\Omega } \bigg [ \, f^{m} ( x ) + h( x ) + \phi ( x ) \cdot \frac{l( u^{m} )}{\bar{a}} \, \bigg ] \cdot \exp ( G( u^{m} ) ) \cdot T_{1} ( u^{m} - T_{j} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,|) \,\, dx \end{aligned}$$

since $\phi \in L^{1} ( \Omega ), \, h \in L^{1} ( \Omega ), \, f^{m} \in ( L^{1} ( \Omega ) )^{N},$ and the fact that $\exp ( G( \pm ) ) \le \exp \bigg ( \frac{||\, l\,||_{L^{1} ( \mathbb {R} )}}{\bar{a}} \bigg )$, we deduce from vitali’s Theorem that

$$\begin{aligned} \lim _{j \rightarrow \infty } \,&\lim _{m \rightarrow \infty } \int _{\Omega } \bigg [ \, f^{m} ( x ) + h( x ) + \phi ( x ) \cdot \frac{l( u^{m} )}{\bar{a}} \, \bigg ] \cdot \exp ( G( u^{m} ) )\cdot T_{1} ( u^{m} - T_{j} ( u^{m} ) ) \\& \times \eta _{R} ( |\, x\,|) \,\, dx = 0. \end{aligned}$$

Hence,

$$\begin{aligned} \lim _{j \rightarrow \infty } \, \lim _{m \rightarrow \infty } \int _{\{\, j< |\, u^{m} \,| < j +1 \,\}} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \eta _{R} ( |\, x\,| )\,\, dx = 0. \end{aligned}$$

And to show that assertion 2 is true, we take

$$\begin{aligned} v = \exp ( G( u^{m} ) ) \,( T_{k} ( u^{m} ) - T_{k} ( u ) ) \, h_{j} ( u^{m} ) \, \eta _{R} ( |\, x\,| ), \end{aligned}$$

as a test function in the problem $(\mathcal {P}_{m} )$. We have

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla \big (\, \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| ) \, \big ) \,\,dx \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega }b_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) )\cdot h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx \\&\quad \le \int _{\Omega } f^{m} ( x ) \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx, \end{aligned}$$

which implies

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \frac{l( u^{m} )}{\bar{a}} \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \\&\quad \times \, \eta _{R} ( |\, x\,| ) \,\, dx \\&\quad +\, \sum _{i = 1}^{N} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot ( \nabla T_{k} ( u^{m} ) - \nabla T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx \\&\quad +\, \sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} )\cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot \nabla h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| )\,\, dx \\&\quad +\, \sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \cdot \nabla \eta _{R} ( |\, x\,| )\,\, dx \end{aligned}$$

$$\begin{aligned}&\le \sum _{i = 1}^{N} \int _{\Omega } |\, b_{i}^{m} ( x, u^{m}, \nabla u^{m} )\,|\cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| ) \,\,dx \\&\quad +\, \int _{\Omega } f^{m} ( x ) \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| ) \,\,dx, \end{aligned}$$

thanks to (22) and (23), we obtain

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot ( \nabla T_{k} ( u^{m} ) - \nabla T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx \\&\quad +\, \sum _{i = 1}^{N} \int _{\{\, \Omega :\, j \le |\,u^{m}\,| \le j + 1 \,\}} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \exp ( G( u^{m} ) ) \\&\quad \times \, ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot \eta _{R} ( |\, x\,| ) \,\,dx \\&\quad + \, \sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \\&\quad \times \, \nabla \eta _{R} ( |\, x\,| ) \,\, dx \\ &\le \int _{\Omega } \bigg [ \, f^{m} ( x ) + h( x ) + \phi ( x ) \cdot \frac{l( u^{m} )}{\bar{a}} \, \bigg ] \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot h_{j} ( u^{m} ) \\&\quad \times \, \eta _{R} ( |\, x\,| )\,\,dx \end{aligned}$$

sine $h_{j} \ge 0$, $\eta _{R} ( |\,x\,| ) \ge 0$ and $u^{m} \,( T_{k} ( u^{m} ) - T_{k} ( u ) ) \ge 0$ we have

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\{\, \Omega :\, |\, u^{m}\,| \le k \, \}} a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u^{m} ) )\exp ( G( u^{m} ) ) \cdot ( \nabla T_{k} ( u^{m} ) - \nabla T_{k} ( u ) ) \\&\qquad \times \, \eta _{R} ( |\, x\,| ) \,\, dx \\&\qquad +\, \int _{\{\, \Omega : \, j \le |\, u^{m}\,| \le j + 1 \,\}} a_{i}^{m} ( x, u^{m}, \nabla u^{m} )\, \nabla u^{m} \, \exp ( G( u^{m} ) )\, ( T_{k} ( u^{m} ) - T_{k} ( u ) )\, \eta _{R} ( |\, x\,| )\, dx \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot \nabla \eta _{R} ( |\, x\,| )\,\, dx\\&\quad \le \int _{\Omega } \bigg [ \, f^{m} ( x ) + h( x ) + \phi ( x ) \cdot \frac{l( u^{m} )}{\bar{a}} \, \bigg ] \cdot \exp ( G( u^{m} ) ) \cdot ( T_{k} ( u^{m} ) - T_{k} ( u ) ) \cdot \eta _{R} ( |\, x\,| )\,\,dx \\&\qquad + \sum _{i = 1}^{N} \int _{\{\, \Omega :\, k \le |\, u^{m}\,| \le j + 1\,\}} a_{i} ( x, T_{j + 1} ( u^{m} ), \nabla T_{j + 1} ( u^{m} ) ) \cdot \exp ( G( u^{m} ) ) \cdot | \, \nabla T_{k} ( u ) \,| \\&\qquad \times \, \eta _{R} ( |\, x\,|) \,\, dx \\&\quad +\, \sum _{i = 1}^{N} \int _{\{ \, \Omega : j \le |\, u^{m} \,| \le j + 1\,\}} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m}\cdot \exp ( G( u^{m} ) ) \cdot |\, T_{k} ( u^{m} ) - T_{k} ( u ) \, | \\&\qquad \times \eta _{R} ( |\, x\,| ) \,\, dx. \end{aligned}$$

The first term in the right hand side goes to zero as m tend to $\infty$, since $T_{k} ( u^{m} ) \rightharpoonup T_{k} ( u )$ weakly in $\mathring{W}_{B}^{1} ( \Omega ( m ) )$.

Since $a_{i}^{m} ( x, T_{j + 1} ( u^{m} ), \nabla T_{j + 1} ( u^{m} ) )$ is bounded in $L_{\bar{B}} ( \Omega ( m ) )$, there exists $\tilde{a}^{m} \in L_{\bar{B}} ( \Omega ( m ) )$ such as

$$\begin{aligned} |\, a_{i}^{m} ( x, T_{j + 1} ( u^{m} ), \nabla T_{j + 1} ( u^{m} ) ) \,| \rightharpoonup \tilde{a}^{m} \,\, \text{ in } \,\, L_{\bar{B}} ( \Omega ( m ) ). \end{aligned}$$

(30)

Thus, the second term of the right hand side goes also to zero.

Since $T_{k} ( u^{m} ) \longrightarrow T_{K} ( u )$ strongly in $\mathring{W}_{B, loc}^{1} ( \Omega ( m ) )$. The third term of the left hand side increased by a quantity that tends to zero as m tend to zero, and according to (28) we deduce that

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\{\, \Omega :\, |\, u^{m}\,| \le k \, \}} a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u^{m} ) ) \cdot \exp ( G( u^{m} ) ) \cdot |\, \nabla T_{k} ( u^{m} ) - \nabla T_{k} ( u ) \,| \\&\qquad \times \, \eta _{R} ( |\, x\,|) \,\, dx\\&\quad \le \epsilon ( j, \, m ). \end{aligned}$$

Then,

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } \bigg [ \, a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u^{m} ) ) - a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u ) ) \, \bigg ] \cdot ( \, \nabla T_{k} ( u^{m} ) - T_{K} ( u ) \, ) \nonumber \\&\qquad \times \eta _{R} ( |\, x\,|) \,\, dx \nonumber \\&\quad \le - \, \sum _{i = 1}^{N} \int _{\Omega } a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u ) ) \cdot \exp ( G( u^{m} ) ) \cdot |\, \nabla T_{k} ( u^{m} ) - \nabla T_{k} ( u ) \,| \nonumber \\&\qquad \times \, \eta _{R} ( |\,x\,|)\,\, dx \nonumber \\&\qquad - \, \sum _{i = 1}^{N} \int _{\{\, \Omega : \, |\, u^{m}\,| \le k \, \}} a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u^{m} ) ) \cdot \exp ( G( u^{m} ) ) \cdot \nabla T_{k} ( u ) \cdot \eta _{R} (|\, x\,|) \,\, dx \nonumber \\&\qquad + \,\epsilon ( j, \, m ). \end{aligned}$$

(31)

According to Lebesgue dominated convergence Theorem, we have $T_{k} ( u^{m} ) \longrightarrow T_{k} ( u )$ in $\mathring{W}_{B, loc}^{1} ( \Omega )$ and $\nabla T_{k} ( u^{m} ) \rightharpoonup \nabla T_{k} ( u )$ in $\mathring{W}_{B}^{1} ( \Omega )$, then the terms on the right had side of (31) goes to zero as m and j tend to infinity. Which implies that

$$\begin{aligned} \sum _{i = 1}^{N}&\int _{\Omega } \bigg [ \, a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u^{m} ) ) - a_{i} ( x, T_{k} ( u^{m} ), \nabla T_{k} ( u ) ) \, \bigg ] \nonumber \\ &\quad \times ( \, \nabla T_{k} ( u^{m} ) - T_{K} ( u ) \, )\,\, dx \longrightarrow 0. \end{aligned}$$

(32)

Thanks to Lemma 2.9, we have for $k = 1, \ldots ,$

$$\begin{aligned} \nabla T_{k} ( u^{m} ) \longrightarrow \nabla T_{k} ( u ) \,\, \text{ a.e } \text{ in } \,\, \Omega ( m ) \end{aligned}$$

(33)

and by diagonal process, we prove that

$$\begin{aligned} \nabla u^{m} \longrightarrow \nabla u \,\, \text{ a.e } \text{ in } \,\, \Omega ( m ). \end{aligned}$$

$\square$

Step 5 Equi-integrability of $b^{m} ( x, u^{m}, \nabla u^{m} )$.

Let $v = \exp ( 2 \, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \eta _{R} ( |\, x \,| )$ as a test function in the problem $( \mathcal {P}_{m} )$, we obtain

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla \big ( \, \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,| ) \, \big ) \,\, dx\\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } b_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx \\&\quad \le \int _{\Omega } f^{m} ( x ) \cdot \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx, \end{aligned}$$

which implies that

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \frac{l( u^{m} )}{\bar{a}} \cdot \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \\&\qquad \times \, \eta _{R} ( |\, x\,| ) \,\, dx \\&\qquad +\, \sum _{i = 1}^{N} \int _{\{ \, \Omega : \, R \le |\, u^{m}\,| \le R + 1 \,\}} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \nabla u^{m} \cdot \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( 2\, G( |\, u^{m}\,| ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \nabla \eta _{R} ( |\, x\,| )\,\, dx \\&\quad \le \sum _{i = 1}^{N} \int _{\Omega } |\,b_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \,| \cdot \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx \\&\qquad +\, \int _{\Omega } f^{m} ( x ) \cdot \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \eta _{R} ( |\, x\,| ) \,\, dx, \end{aligned}$$

by (22) and (23), we obtain

$$\begin{aligned}&\bar{a} \, \sum _{i = 1}^{N} \int _{\{ \, \Omega : \, R \le |\, u^{m}\,| \le R + 1 \,\}} B_{i} ( |\, \nabla u^{m} \,| ) \cdot \exp ( 2\, G( |\, u^{m} \,| ) \cdot \eta _{R} ( |\, x \,| ) \,\, dx \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \exp ( 2\, G( |\, u^{m}\,| ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \cdot \nabla \eta _{R} ( |\, x\,| )\,\, dx \\&\quad \le \int _{\Omega } \bigg [ \, f^{m} ( x ) + h( x ) + \phi ( x ) \cdot \frac{l( u^{m} )}{\bar{a}} \, \bigg ] \cdot \exp ( 2\, G( |\,u^{m}\,| ) ) \cdot T_{1} ( u^{m} - T_{R} ( u^{m} ) ) \\ &\quad \times \eta _{R} ( |\, x\,| ) \,\, dx + \int _{\{ \, \Omega : \, R \le |\, u^{m}\,| \le R + 1 \,\}} \phi ( x ) \cdot \exp ( 2\, G( |\, u^{m} \,| ) \cdot \eta _{R} ( |\, x\,| ) \,\,dx. \end{aligned}$$

Since $\eta _{R} ( |\, x\,|) \ge 0, \,\, \exp ( G( \pm \infty ) ) \le \exp \bigg ( \, 2 \,\frac{||\, l\,||_{L^{1}} ( \mathbb {R} )}{\bar{a}} \, \bigg ), \, f^{m} \in ( \, L^{1} ( \Omega ) \,)^{N}, \, \phi$ and $h \, \in L^{1} ( \Omega )$. Then, $\forall \, \epsilon> 0, \,\, \exists \, R( \epsilon ) > 0$ such as

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\{\,\Omega : \, |\, u^{m} \,|> R + 1 \,\}} B( |\, \nabla u^{m} \,| ) \,\, dx \le \frac{\epsilon }{2} \quad \forall R > R( \epsilon ). \end{aligned}$$

Let $\mathring{V} ( \Omega ( m ) )$ be an arbitrary bounded subset for $\Omega$, then, for any measurable set $E \subset \mathring{V} ( \Omega ( m ) )$ we have

$$\begin{aligned} \sum _{i = 1}^{N} \int _{E} B_{i} ( |\, \nabla u^{m} \,| )\,\, dx&\le \sum _{i = 1}^{N} \int _{E} B_{i} ( |\, \nabla T_{R} ( u^{m} ) \, | ) \,\, dx \nonumber \\&\quad +\, \sum _{i = 1}^{N} \int _{\{\, |\, u^{m} \,| > R + 1 \,\}} B_{i} ( |\, \nabla u^{m} \,| ) \,\, dx \end{aligned}$$

(34)

we conclude that $\forall E \subset \mathring{V} ( \Omega ( m ) )$ with $\text{ meas } \, ( E ) < \beta ( \epsilon )$ and $T_{R} ( u^{m} ) \longrightarrow T_{R} ( u )$ in $\mathring{W}_{B}^{1} ( \Omega )$

$$\begin{aligned} \sum _{i = 1}^{N} \int _{E} B_{i} ( |\, \nabla T_{R} ( u^{m} ) \, | ) \,\, dx \le \frac{\epsilon }{2}. \end{aligned}$$

(35)

Finally, according to (34) and (35), we obtain

$$\begin{aligned} \sum _{i = 1}^{N} \int _{E} B_{i} ( |\, \nabla u^{m} \,| ) \,\, dx \le \epsilon \quad \forall E \subset \mathring{V} ( \Omega ( m ) ) \,\, \text{ such } \text{ as } \,\, \text{ meas } \, ( E ) < \beta ( \epsilon ). \end{aligned}$$

Which gives the results.

Step 6 Passing to the limit.

Let $\xi \in \mathring{W}_{B}^{1} ( \Omega ) \cap L^{\infty } ( \Omega )$, using the following test function $v = \vartheta _{k} \, T_{k} ( u^{m} - \xi )$ in the problem $( \mathcal {P}_{m} )$ with

$$\begin{aligned} \vartheta _{k} = {\left\{ \begin{array}{ll} 1 &\quad \text {for} \,\,\Omega ( m ) , \\ 0 &\quad \text {for} \, \Omega ( m + 1 ) \backslash \Omega ( m ) . \end{array}\right. } \end{aligned}$$

and $|\, u^{m}\,| - ||\, \xi \,||_{\infty } < |\, u^{m} - \xi \,| \le j$. Then, $\{\, |\, u^{m} - \xi \,| \le j \, \} \subset \{\, |\, u^{m}\,| \le j + ||\, \xi \,||_{\infty } \, \}$ we obtain

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } a_{i} ( x, T_{m} ( u^{m} ), \nabla u^{m} )\cdot \vartheta _{k} \, \nabla T_{k} ( u^{m} - \xi ) \,\, dx\nonumber \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } a_{i} ( x, T_{m} ( u^{m} ), \nabla u^{m} ) \cdot T_{k} ( u^{m} - \xi ) \, \nabla \vartheta _{k} \,\, dx \nonumber \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } b_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \vartheta _{k} \, T_{k} ( u^{m} - \xi ) \,\, dx \nonumber \\&\quad \le \int _{\Omega } f^{m} ( x ) \cdot \vartheta _{k} \,T_{k} ( u^{m} - \xi ) \,\, dx \end{aligned}$$

(36)

which implies that

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega ( m )} a_{i} ( x, T_{m} ( u^{m} ), \nabla u^{m} ) \cdot T_{k} ( u^{m} - \xi ) \,\, dx \nonumber \\&\quad = \sum _{i = 1}^{N} \int _{\Omega ( m )} a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ), \nabla T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ) )\cdot T_{j + ||\, \xi \,||_{\infty }} ( u^{m} - \xi ) \nonumber \cdot \chi _{\{\, |\, u^{m} - \xi \,|< j \,\}} dx \nonumber \\&\quad = \sum _{i = 1}^{N} \int _{\Omega ( m )} \bigg [ \, a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ), \nabla T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ) ) - a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ), \nabla \xi ) \, \bigg ] \nonumber \\&\qquad \times \, \nabla T_{j + ||\, \xi \,||_{\infty }} ( u^{m} - \xi ) \cdot \chi _{\{\, |\, u^{m} - \xi \,|< j \,\}} \,\ dx \nonumber \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega ( m )} a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ), \nabla \xi ) \cdot \nabla T_{j + ||\, \xi \,||_{\infty }} ( u^{m} - \xi ) \cdot \chi _{\{\, |\, u^{m} - \xi \,| < j \,\}} \,\ dx. \end{aligned}$$

(37)

By Fatou’s Lemma, we have

$$\begin{aligned}&\lim _{m \rightarrow \infty } \inf \, \sum _{i = 1}^{N} \int _{\Omega ( m )} a_{i} ( x, T_{m} ( u^{m} ), \nabla u^{m} ) \cdot \nabla T_{k} ( u^{m} - \xi ) \,\, dx\nonumber \\&\quad \ge \, \sum _{i = 1}^{N} \int _{\Omega ( m )} \bigg [ \, a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ), \nabla T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ) ) - a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ), \nabla \xi ) \, \bigg ]\nonumber \\&\qquad \times \, \nabla T_{j + ||\, \xi \,||_{\infty }} ( u^{m} - \xi ) \cdot \chi _{\{\, |\, u^{m} - \xi \,|< j \,\}} \,\ dx \nonumber \\&\quad + \lim _{m \rightarrow \infty } \sum _{i = 1}^{N} \int _{\Omega ( m )} a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u^{m} ), \nabla \xi ). \nabla T_{j + ||\, \xi \,||_{\infty }} ( u^{m} - \xi )\cdot \chi _{\{\, |\, u^{m} - \xi \,| < j \,\}} \,\ dx. \end{aligned}$$

(38)

The second term on the right hand side of the previous inequality is equal to

$$\begin{aligned} \int _{\Omega ( m )} a_{i} ( x, T_{j + ||\, \xi \,||_{\infty }} ( u ), \nabla \xi ) \cdot \nabla T_{j + ||\, \xi \,||_{\infty }} ( u - \xi ) \cdot \chi _{\{\, |\, u - \xi \,| < j \,\}} \,\ dx. \end{aligned}$$

Then, since $T_{k} ( u^{m} - \xi ) \rightharpoonup T_{k} ( u - \xi ) \,\, \text{ weakly } \text{ in } \,\, \mathring{W}_{B}^{1} ( \Omega )$, and by (29), (33) we have

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\Omega } b_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot \vartheta _{k} \, T_{k} ( u^{m} - \xi ) \,\, dx \longrightarrow \sum _{i = 1}^{N} \int _{\Omega } b_{i} ( x, u, \nabla u ) \cdot \vartheta _{k} \, T_{k} ( u - \xi ) \,\, dx \end{aligned}$$

(39)

and

$$\begin{aligned} \int _{\Omega } f^{m} ( x ) \cdot \vartheta _{k} \, T_{k} ( u^{m} - \xi ) \,\, dx \longrightarrow \int _{\Omega } f ( x ) \cdot \vartheta _{k} \, T_{k} ( u - \xi ) \,\, dx. \end{aligned}$$

(40)

Combining (36)–(40) and passing to the limit as $m \longrightarrow \infty$, we have the condition 3 in Definition 1.1. $\square$

4 Uniqueness result in unbounded domain

In this section, we demonstrate the Theorem of uniqueness to the solution of problem $( \mathcal {P} )$ in an unbounded domain; using the the fact given in [1, 11, 12] such as $b_{i} ( x, u, \nabla u )$ are a contraction Lipschitz continuous functions.

Theorem 4.1

Under assumptions (20)–(23), and $b_{i} ( x, u, \nabla u ): \Omega \times \mathbb {R} \times \mathbb {R}^{N} \longrightarrow \mathbb {R}$ for $i = 1, \ldots , N$ contraction Lipschitz continuous functions do not satisfy any sign condition, and

$$\begin{aligned} \sum _{i = 1}^{N} \big [ \, a_{i} ( x, \xi , \nabla \xi ) - a_{i} ( x, \xi ^{'}, \nabla \xi ^{'} ) \, \big ] \cdot ( \nabla \xi - \nabla \xi ^{'} ) > 0. \end{aligned}$$

(41)

The problem $( \mathcal {P} )$ has a unique solution.

Proof

Let $u^{1}$ and $u^{2}$ be two solutions of problem $( \mathcal {P} )$ with $u^{1} \ne u^{2}$ then,

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\Omega } a_{i} ( x, u^{1}, \nabla u^{1} ) \cdot \nabla v \,\, dx + \sum _{i = 1}^{N} \int _{\Omega } b_{i} ( x, u^{1}, \nabla u^{1} ) \cdot v \,\, dx = \int _{\Omega } f( x ) \cdot v \,\,dx \end{aligned}$$

and

$$\begin{aligned} \sum _{i = 1}^{N} \int _{\Omega } a_{i} ( x, u^{2}, \nabla u^{2} ) \cdot \nabla v \,\, dx + \sum _{i = 1}^{N} \int _{\Omega } b_{i} ( x, u^{2}, \nabla u^{2} ) \cdot v \,\, dx = \int _{\Omega } f( x ) \cdot v \,\,dx \end{aligned}$$

we subtract the previous inequality, we get

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } \big [ \, a_{i} ( x, u^{1}, \nabla u^{1} ) - a_{i} ( x, u^{2}, \nabla u^{2} ) \, \big ] \cdot \nabla v \,\, dx \\&\quad +\, \sum _{i = 1}^{N} \int _{\Omega } \big [ \, b_{i} ( x, u^{1}, \nabla u^{1} ) - b_{i} ( x, u^{2}, \nabla u^{2} ) \, \big ] \cdot v \,\, dx = 0 \end{aligned}$$

we take $v = \eta ( x ) \cdot ( u^{1} - u^{2} ) ( x )$ with

$$\begin{aligned} \eta ( x )= {\left\{ \begin{array}{ll} 0 &\quad \text {if} \,\,x \ge k , \\ k - \frac{|\, x\,|^{2}}{k} &\quad \text {if} \, |\, x\,| < k , \\ 0 &\quad \text {if} \,\,x \le - k . \end{array}\right. } \end{aligned}$$

Combine to (41), we obtain

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } \big [ \, a_{i} ( x, u^{1}, \nabla u^{1} ) - a_{i} ( x,u^{2}, \nabla u^{2} )\, \big ]\cdot ( u^{1} - u^{2} ) \cdot \nabla \eta ( x ) \,\, dx \\&\qquad +\, \sum _{i = 1}^{N} \int _{\Omega } \big [ \, b_{i} ( x, u^{1}, \nabla u^{1} ) - b_{i} ( x,u^{2}, \nabla u^{2} )\, \big ]\cdot ( u^{1} - u^{2} ) \cdot \eta ( x ) \,\, dx \\&\quad \le 0 \end{aligned}$$

according to (2) and the fact that $b_{i} ( x, u, \nabla u )$ contraction Lipschitz functions for $i = 1, \ldots , N$, we get

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} \big ( a_{i} ( x, u^{1}, \nabla u^{1} ) - a_{i} ( x, u^{2}, \nabla u^{2} ) \big ) \, dx + \sum _{i = 1}^{N} \int _{\Omega } B_{i} ( u^{1} - u^{2} ) \, \nabla \eta ( x ) )\, dx \nonumber \\ &\le \sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} \big ( \, a_{i} ( x, u^{1}, \nabla u^{1} ) - a_{i} ( x, u^{2}, \nabla u^{2} ) \, \big ) \,\, dx + 2 \, \sum _{i = 1}^{N} \int _{\Omega } B_{i} ( u^{1} - u^{2} ) \,\, dx \nonumber \\ &\le \alpha \, \sum _{i = 1}^{N} \int _{\Omega } B_{i} ( u^{1} - u^{2} ) \,\, dx + \alpha \, \sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} ( \eta ( x ) \cdot ( u^{1} - u^{2} ) ) \,\, dx \end{aligned}$$

(42)

then

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} \big ( \, a_{i} ( x, u^{1}, \nabla u^{1} ) - a_{i} ( x, u^{2}, \nabla u^{2} ) \, \big ) \,\, dx \nonumber \\&\quad \le ( \alpha - 2 ) \, \sum _{i = 1}^{N} \int _{\Omega } B_{i} ( u^{1} - u^{2} ) \,\, dx + \alpha \, \sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} ( \eta ( x ) \cdot ( u^{1} - u^{2} ) ) \,\, dx. \end{aligned}$$

(43)

Since,

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} ( \eta ( x ) \cdot ( u^{1} - u^{2} ) ) \,\, dx\\ &\le \sum _{i = 1}^{N} \int _{\Omega \cap \{\, |\, x\,| \le k \,\}} \bar{B}_{i} \bigg ( \,\bigg ( \, k - \frac{|\, x\,|^{2}}{k} \, \bigg ) \cdot ( u^{1} - u^{2} ) \, \bigg ) \,\, dx \\&\quad +\, \sum _{i = 1}^{N} \int _{\Omega \cap \{\, |\, x\,| > k \,\}} \bar{B}_{i} ( \, \eta ( x ) \cdot ( u^{1} - u^{2} ) )\,\, dx\\&\longrightarrow 0 \,\, \text{ as }\,\, k \longrightarrow 0 \end{aligned}$$

and since the N-functions $\bar{B}_{i}$ verified the same conditions and properties of the $B_{i}$ then, according to (6) and (20), we obtain

$$\begin{aligned}&\sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} \big ( \, a_{i} ( x, u^{1}, \nabla u^{1} ) - a_{i} ( x, u^{2}, \nabla u^{2} ) \, \big ) \,\, dx \\&\quad \le \tilde{a} c \, \sum _{i = 1}^{N} \int _{\Omega } B_{i} ( \nabla ( u^{2} - u^{2} ) ) \,\, dx \\&\le \tilde{a} c \, ||\, B( u^{1} - u^{2} ) \,||_{1, \Omega }. \end{aligned}$$

Combine to (42) and (43), we deduce that

$$\begin{aligned} 0 \le ( \tilde{a} c + 2 - \alpha ) \, ||\, B( u^{1} - u^{2} ) \,||_{1, \Omega } \le 0. \end{aligned}$$

Thus

$$\begin{aligned} ||\, B( u^{1} - u^{2} ) \,||_{1, \Omega } = 0. \end{aligned}$$

Hence, $u^{1} = u^{2}$ a.e in $\Omega$. $\square$

Appendix

Let

$$\begin{aligned}&A : \mathring{W}_{B}^{1} ( \Omega ) \longrightarrow ( \mathring{W}_{B}^{1} ( \Omega ) )^{'} \\&\quad v \quad \longmapsto < A( u ), \, v > = \int _{\Omega } \sum _{i = 1}^{N} \bigg ( \, a_{i} ( x, u, \nabla u ) \cdot \frac{\partial v}{\partial x_{i}} + b_{i} ( x, u, \nabla u ) \cdot v \,\bigg ) \,\, dx \\&\quad -\, \int _{\Omega } f( x )\cdot v \,\, dx \end{aligned}$$

and let denote $L_{\bar{B}} ( \Omega ) = \displaystyle \prod _{k=1}^{N} L_{\bar{B}_{i}} ( \Omega )$ with the norm

$$\begin{aligned} ||\, v\,||_{L_{\bar{B}} ( \Omega )} = \sum _{i = 1}^{N} ||\, v_{i} \,||_{\bar{B}_{i}, \Omega } \quad v = ( v_{1}, \ldots , v_{N} ) \in L_{\bar{B}} ( \Omega ). \end{aligned}$$

Where $\bar{B_{i}} ( t )$ are N-functions satisfying the $\Delta _{2}-$conditions. Sobolev-space $\mathring{W}_{B}^{1} ( \Omega )$ is the completions of the space $C_{0}^{\infty } ( \Omega )$.

$$\begin{aligned} a( x, s, \xi ) = \big ( \, a_{1} ( x, s, \xi ), \ldots , a_{N} ( x, s, \xi ) \, \big ) \end{aligned}$$

and

$$\begin{aligned} b( x, s, \xi ) = \big ( \, b_{1} ( x, s, \xi ), \ldots , b_{N} ( x, s, \xi ) \, \big ). \end{aligned}$$

Let’s show that operator A is bounded, so for $u \in \mathring{W}_{B}^{1} ( \Omega )$, according to (9) and (20) we get

$$\begin{aligned} || \,a( x, u, \nabla u ) \,||_{L_{\bar{B}} ( \Omega )}&= \sum _{i = 1}^{N} ||\, a_{i} ( x, u, \nabla u ) \, ||_{L_{\bar{B}_{i}} ( \Omega )} \nonumber \\&\le \sum _{i = 1}^{N} \int _{\Omega } \bar{B}_{i} ( \, a_{i} ( x, u, \nabla u )\, ) \,\, dx + N \nonumber \\ &\le \tilde{a} ( \Omega ) \cdot || \, B( u ) \, ||_{1, \Omega } + ||\, \varphi ||_{1, \Omega } + N. \end{aligned}$$

(44)

Further, for $a( x, u, \nabla u ) \in L_{\bar{B}_{i}} ( \Omega ), \,\,\, v \in \mathring{W}_{B}^{1} ( \Omega )$ using Hölder’s inequality we have

$$\begin{aligned} |\,< A( u ), \, v >_{\Omega } \, |&\le \,2 \, || \, a( x, u, \nabla u ) \, ||_{L_{\bar{B}} ( \Omega )} \cdot ||\, v \,||_{\mathring{W}_{B}^{1} ( \Omega )} \nonumber \\&\quad +\, 2 \, ||\, b( x, u, \nabla u ) \,||_{L_{B} ( \Omega )} \cdot ||\, v\,||_{\mathring{W}_{B}^{1} ( \Omega )} + c_{0} \cdot ||\, v\,||_{\mathring{W}_{B}^{1} ( \Omega )}. \end{aligned}$$

(45)

Thus, A is bounded. And that A is coercive, so for $u \in \mathring{W}_{B}^{1} ( \Omega )$

$$\begin{aligned} < A( u ), u >_{\Omega }&= \sum _{i = 1}^{N} \int _{\Omega } a_{i} ( x, u, \nabla u ) \cdot \frac{\partial u}{\partial x_{i}} \,\, dx \,+ \, \sum _{i = 1}^{N} \int _{\Omega } b_{i} ( x, u, \nabla u ) \cdot u \,\, dx\\&\quad -\, \int _{\Omega } f( x ) \cdot u \,\, dx. \end{aligned}$$

Then,

$$\begin{aligned} \frac{< A( u ), \, u >_{\Omega }}{||\, u\,||_{\mathring{W}_{B}^{1} ( \Omega )}}&\ge \frac{1}{||\, u\,||_{\mathring{W}_{B}^{1} ( \Omega )}} \cdot \bigg [ \, \bar{a} \, \sum _{i = 1}^{N} \int _{\Omega } B_{i} \bigg ( \,\bigg |\, \frac{\partial u}{\partial x_{i}} \,\bigg |\, \bigg ) \,\, dx - c_{1} - c_{0} \\&\qquad -\, l( u ) \cdot \sum _{i = 1}^{N} \int _{\Omega } B_{i} \bigg ( \,\bigg |\, \frac{\partial u}{\partial x_{i}} \,\bigg |\, \bigg ) \,\, dx - \int _{\Omega } h( x )\,\, dx \, \bigg ] \\&\quad \ge \frac{1}{||\, u\,||_{\mathring{W}_{B}^{1} ( \Omega )}} \cdot \bigg [ \, ( \, \bar{a} ( \Omega ) - c_{2} \,) \cdot \sum _{i = 1}^{N} \int _{\Omega } B_{i} \bigg ( \,\bigg |\, \frac{\partial u}{\partial x_{i}} \,\bigg |\, \bigg ) \,\, dx - c_{0} - c_{1}- c_{3} \, \bigg ] \\ \end{aligned}$$

According to (20), we have for all $k> 0, \,\, \exists \, \alpha _{0} > 0$ such that

$$\begin{aligned} b_{i} ( \, |\, u_{x_{i}} \,| \, ) > k \, b_{i} \bigg ( \, \frac{|\, u_{x_{i}} \, |}{||\, u_{x_{i}}\,||_{B_{i}, \Omega }} \, \bigg ) , \quad i = 1, \ldots , N. \end{aligned}$$

We take $||\, u_{x_{i}} \,||_{B_{i}, \Omega } > \alpha _{0} \quad i = 1, \ldots , N.$

Suppose that $||\, u_{x_{i}} \, ||_{\mathring{W}_{B}^{1} ( \Omega )} \longrightarrow 0$ as $j \rightarrow \infty$. We can assume that

$$\begin{aligned} || \, u^{j}_{x_{1}} \, ||_{B_{1}, \Omega } + \cdots + || \, u^{j}_{x_{N}}||_{B_{N}, \Omega } \ge N \, \alpha _{0}. \end{aligned}$$

According to (9) for $c > 1,$ we have

$$\begin{aligned} |\, u^{j}\,| \, b( \, |\, u^{j}\,|\,) < c \, B( u^{j} ) \end{aligned}$$

then, by (2.8) we obtain

$$\begin{aligned} \frac{< A( u^{j} ), \, u^{j} >_{\Omega }}{||\, u^{j}\,||_{\mathring{W}_{B}^{1} ( \Omega )}}&\ge \frac{\bar{a} ( \Omega ) - c_{2}}{N \, \alpha _{0}} \cdot \sum _{i = 1}^{N} \int _{\Omega } B_{i} \bigg ( \,\bigg |\, \frac{\partial u}{\partial x_{i}} \,\bigg |\, \bigg ) \,\, dx - \frac{c_{4}}{N \, \alpha _{0}} \\&\ge \frac{\bar{a} ( \Omega ) - c_{2}}{N \, \alpha _{0}} \cdot \sum _{i = 1}^{N} \int _{\Omega } |\, u^{j}_{x_{i}}\,| \, b( \, |\, u^{j}_{x_{i}}\,|\,) \,\, dx - \frac{c_{4}}{N \, \alpha _{0}} \\&\ge \frac{( \,\bar{a} ( \Omega ) - c_{2} \, ) \cdot k}{c\, N \, ||\, u^{j}_{x_{i}} \, ||_{B_{i}}} \cdot \sum _{i = 1}^{N} \int _{\Omega } |\, u^{j}_{x_{i}}\,| \, b_{i} \bigg ( \, \frac{|\, u^{j}_{x_{i}} \, |}{||\, u^{j}_{x_{i}}\,||_{B_{i}, \Omega }} \, \bigg ) \,\, dx - \frac{c_{4}}{N \, \alpha _{0}} \\ &\ge \frac{( \,\bar{a} ( \Omega ) - c_{2} \, ) \cdot k}{c\, N} \cdot \sum _{i = 1}^{N} \int _{\Omega } B_{i} \bigg ( \, \frac{ |\, u^{j}_{x_{i}}\,|}{||\, u^{j}_{x_{i}}\,||_{B_{i}, \Omega }} \, \bigg ) \,\, dx - \frac{c_{4}}{N \, \alpha _{0}} \\&\ge \frac{( \,\bar{a} ( \Omega ) - c_{2} \, ) \cdot k}{c\, N} - \frac{c_{4}}{N \, \alpha _{0}}. \end{aligned}$$

which shows that A is coercive, because k is arbitrary.

And for A pseudo-monotonic, we consider a sequence $\{\, u^{m}\,\}_{m = 1}^{\infty }$ in the space $\mathring{W}_{B}^{1} ( \Omega )$ such that

$$\begin{aligned} u^{m} \rightharpoonup u \,\, \text{ weakly } \text{ in } \,\, \mathring{W}_{B}^{1} ( \Omega ) \quad m \rightarrow \infty . \end{aligned}$$

(46)

$$\begin{aligned} \lim _{m \rightarrow \infty } \sup \,< \, A( u^{m} ), \, u^{m} - u \, > \le 0 \end{aligned}$$

(47)

we demonstrate that

$$\begin{aligned}&A( u^{m} ) \rightharpoonup A( u ) \,\, \text{ weakly } \text{ in } \,\, ( \, \mathring{W}_{B}^{1} ( \Omega )\, )^{'}, \,\, m \rightarrow \infty . \end{aligned}$$

(48)

$$\begin{aligned}&< \, A( u^{m} ), \, u^{m} - u \,> \longrightarrow 0, \,\, m \rightarrow \infty . \end{aligned}$$

(49)

Since $B( \theta )$ satisfy the $\Delta _{2}$-condition, then by (9) we have

$$\begin{aligned} \int _{\Omega } B( \theta ) \,\, dx \le c_{0} \, ||\, \theta \,||_{B, \, \Omega }. \end{aligned}$$

(50)

According to (46) we get

$$\begin{aligned} ||\, u^{m} \,||_{\mathring{W}_{B}^{1} ( \Omega )} \le c_{1} \quad m = 1, 2, \ldots \end{aligned}$$

(51)

and

$$\begin{aligned} ||\, B( \nabla u^{m} ) \,||_{1} \le c_{2} \quad m = 1, 2, \ldots . \end{aligned}$$

(52)

Combining to (44) and (51) we obtain

$$\begin{aligned} ||\, a^{m}( x, u, \nabla u )\,||_{\bar{B}} = \sum _{i = 1}^{N} ||\, a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \,||_{\bar{B}_{i}} \le c_{3} \,\, m = 1,2, \ldots . \end{aligned}$$

(53)

And for $m \in \mathbb {N}^{*}, \,\, |\, b^{m} ( x, u, \nabla ) \,| = |\, T_{m} ( b( x, u, \nabla u )\,| \le m.$ Then, by (23) and (51) we have

$$\begin{aligned}&||\, b^{m} ( x, u, \nabla u )\,||_{B} = \sum _{i = 1}^{N} ||\, b_{i}^{m} ( x, u^{m}, \nabla u^{m} )\,||_{B_{i}} \le c_{4} \,\, m = 1,2, \ldots . \end{aligned}$$

According again to proof of Lemmas 3.4 and 2.8, we have

$$\begin{aligned}&\mathring{W}_{B}^{1} ( \Omega ( R + 1 ) ) \hookrightarrow L_{B_{i}} ( \Omega ( R + 1 ) ) \,\, \text{ for } \,\, R > 0 \,\, \text{ and } \,\, i = 1, \ldots , N. \end{aligned}$$

We set

$$\begin{aligned} A^{m} ( x )&= \sum _{i = 1}^{N} \big [ \, a^{m}_{i} ( x, u^{m}, \nabla u^{m} ) - a^{m}_{i} ( x, u, \nabla u ) \, \big ] \, ( u^{m} - u )_{x_{i}} \\&\quad +\, \sum _{i = 1}^{N} \big [ \, b^{m}_{i} ( x, u^{m}, \nabla u^{m} ) - b^{m}_{i} ( x, u, \nabla u ) \, \big ] \, ( u^{m} - u ), \,\, m = 1, \ldots . \end{aligned}$$

then

$$\begin{aligned} < \, A( u^{m} ) - A( u ), \, u^{m} - u \, > = \int _{\Omega } A^{m} ( x ) \,\, dx \quad m = 1, \ldots . \end{aligned}$$

By (46) and (47), we obtain

$$\begin{aligned} \lim _{m \rightarrow \infty } \sup \, \int _{\Omega } A^{m} ( x ) \,\, dx \le 0. \end{aligned}$$

So,

$$\begin{aligned} A^{m} ( x )&= \sum _{i = 1}^{N} \big [ \, a^{m}_{i} ( x, u^{m}, \nabla u^{m} ) - a^{m}_{i} ( x, u^{m}, \nabla u ) \, \big ] \, ( u^{m} - u )_{x_{i}} \nonumber \\&\quad +\, \sum _{i = 1}^{N} \big [ \, a^{m}_{i} ( x, u^{m}, \nabla u ) - a^{m}_{i} ( x, u, \nabla u ) \, \big ] \, ( u^{m} - u )_{x_{i}} \nonumber \\&\quad +\, \sum _{i = 1}^{N} \big [ \, b^{m}_{i} ( x, u^{m}, \nabla u^{m} ) - b^{m}_{i} ( x, u, \nabla u ) \, \big ] \, ( u^{m} - u ) \nonumber \\&= A^{m}_{1} ( x ) + A_{2}^{m} ( x ) + A_{3}^{m} ( x ) \quad m = 1, \ldots . \end{aligned}$$

(54)

We prove that

$$\begin{aligned} A_{1}^{m} ( x ) \longrightarrow 0 \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega \quad m \rightarrow \infty . \end{aligned}$$

(55)

$$\begin{aligned} A_{2}^{m} ( x ) \longrightarrow 0 \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega \quad m \rightarrow \infty . \end{aligned}$$

(56)

$$\begin{aligned} A_{3}^{m} ( x ) \longrightarrow 0 \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega \quad m \rightarrow \infty . \end{aligned}$$

(57)

$$\begin{aligned} A^{m} ( x )&= \sum _{i = 1}^{N} \big [ \, a^{m}_{i} ( x, u^{m}, \nabla u^{m} ) - a^{m}_{i} ( x, u^{m}, \nabla u ) \, \big ] \, ( u^{m} - u )_{x_{i}} \\&= \sum _{i = 1}^{N} a_{i}^{m} ( x, u^{m}, \nabla u^{m} )\cdot u_{x_{i}}^{m} - \sum _{i = 1}^{N} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \cdot u_{x_{i}} \\&\quad -\, \sum _{i = 1}^{N} a_{i}^{m} ( x, u, \nabla u ) \cdot u^{m}_{x_{i}} + \sum _{i = 1}^{N} a_{i}^{m} ( x, u, \nabla u ) \cdot u_{x_{i}} \end{aligned}$$

applying (1), (22), (52) and (53) we obtain

$$\begin{aligned} A^{m}_{1} ( x ) \ge c( m ) \longrightarrow 0 \,\, \text{ as } \,\, m \rightarrow \infty . \end{aligned}$$

Hence, using the diagonal process, we conclude the convergence (55).

As in [32], let $A_{i} ( u ) = a_{i} ( x, u, \nabla v ) \,\, i = 1, \ldots , N$ be Nemytsky operators for $v \in \mathring{W}_{B}^{1} ( \Omega )$ fixed and $x \in \Omega ( R ),$ continuous in $L_{\bar{B}_{i}} ( \Omega ( R ) )$ for any $R > 0.$

Thus, according to (10), (27) and the diagonal process, we have for any $R > 0$

$$\begin{aligned} A_{2}^{m} ( x ) \longrightarrow 0 \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega \quad m \rightarrow \infty . \end{aligned}$$

Applying the inequality (10) we obtain

$$\begin{aligned} A_{3}^{m} ( x )&\le 2 \, \sum _{i = 1}^{N} ||\, b_{i}^{m} ( x, u^{m}, \nabla u^{m} ) - b_{i}^{m} ( x, u, \nabla u ) \,||_{B_{i}, \Omega ( R )} \cdot ||\, u^{m} - u \,||_{\mathring{W}_{B}^{1} ( \Omega )} \\ &\le 2 c( m ) \cdot ||\, u^{m} - u \,||_{\mathring{W}_{B}^{1} ( \Omega )}. \end{aligned}$$

Hence, combining to (27) and the diagonal process, we have for any $R > 0$

$$\begin{aligned} A_{3}^{m} ( x ) \longrightarrow 0 \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega \quad m \rightarrow \infty . \end{aligned}$$

Consequently, by (55), (56), (57) and the selective convergences we deduce that

$$\begin{aligned} A^{m} ( x ) \longrightarrow 0 \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega \quad m \rightarrow \infty . \end{aligned}$$

(58)

Let $\Omega ' \subset \Omega , \, \text{ meas }\, \Omega ' = \text{ meas } \,\Omega$, and the conditions (27), (58) are true, and (20)–(23) are satisfied.

We prove the convergence

$$\begin{aligned} u^{m}_{x_{i}} ( x ) \, \longrightarrow \, u_{x_{i}} ( x ) \,\, \text{ everywhere } \text{ in } \,\, \Omega \,\,\text{ for }\,\, i = 1, \ldots , N \,\, , \,\, m \rightarrow \infty \end{aligned}$$

(59)

By the absurd, suppose we do not have convergence at the point $x ^ {*} \in \Omega '$.

Let $u^{m} \, = u^{m}_{x_{i}} ( x^{*} ) , \, u = u_{x_{i}} ( x^{*} ) , \,\, i = 1, \ldots , N,$ and $\hat{a} = \varphi _{1} ( x^{*} ) ,\,\, \bar{a} = \varphi ( x^{*} ) .$

Suppose that the sequence $\,\displaystyle \sum _ {i = 1} ^ {N} B_ {i} (u^{m}) \, \, m = 1, \ldots , \infty$ is unbounded.

Let $\epsilon \in \bigg ( 0 , \frac{\bar{a}}{1 + \hat{a}} \bigg )$ is fixed, according to (2), (4) and the conditions (20), (22), we get

$$\begin{aligned} A^{m} ( x^{*} )&= \sum _{i = 1}^{N} \bigg ( a_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) - a_{i}^{m} ( x^{*}, u, \nabla u ) \bigg ) \, \nabla ( u^{m} - u )\\&\quad +\, \sum _{i = 1}^{N} \bigg ( b_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) - b_{i}^{m} ( x^{*}, u , \nabla u ) \bigg ) \, ( u^{m} - u ) \\&= \sum _{i = 1}^{N} a_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) \, \nabla u^{m} - \sum _{i = 1}^{N} a_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) \, \nabla u \\&\quad -\,\sum _{i = 1}^{N} a_{i}^{m} ( x^{*}, u , \nabla u ) \, \nabla u^{m} + \, \sum _{i = 1}^{N} a_{i}^{m} ( x^{*}, u, \nabla u ) \, \nabla u \\&\quad + \, \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) \, u^{j} \, - \, \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) \, u \\&\quad - \, \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u , \nabla u ) \, u^{m} \, + \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u , \nabla u ) \, u. \end{aligned}$$

Applying the generalized Young inequality and (51), we obtain

$$\begin{aligned} A^{m} ( x^{*} )&\ge \sum _{i = 1}^{N} a_{i}^{m} ( x^{*}, u , \nabla u ) \cdot \nabla u \, + \, \sum _{i = 1}^{N} a_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) \cdot \nabla u^{m} \\&\qquad - \, \epsilon \, \sum _{i = 1}^{N} \bar{B}_{i} ( a_{i}^{m} ( x^{*}, u^{m}, \nabla u^{m} ) ) \\&\qquad - \,c_{1} ( \epsilon ) \,\sum _{i = 1}^{N} B_{i} ( \nabla u ) - \, \epsilon \, \sum _{i = 1}^{N} \bar{B}_{i} ( a_{i}^{m} ( x^{*}, u, \nabla u ) ) - \, c_{2} ( \epsilon ) \, \sum _{i = 1}^{N} B_{i} ( \nabla u^{m} ) \\&\qquad +\, \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) \cdot \nabla u^{m} + \, \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u , \nabla u ) \cdot \nabla u \\&\qquad - \, \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u^{m} , \nabla u^{m} ) \cdot \nabla u \\&\qquad - \, \sum _{i = 1}^{N} b_{i}^{m} ( x^{*}, u , \nabla u ) \cdot \nabla u^{m}\\&\quad \ge \, \bar{a} \, \sum _{i = 1}^{N} B_{i} ( \nabla u ) \, - \, \psi ( x^{*} ) \, + \, \sum _{i = 1}^{N} B_{i} ( \nabla u^{m} ) \, - \, \psi ( x^{*} )\, \\&\qquad - \, \epsilon \, \hat{a} \, \sum _{i = 1}^{N} B_{i} ( \nabla u^{m} ) - \, \epsilon \, \varphi ( x^{*} ) \\&\qquad - \, c_{1} ( \epsilon ) \, \sum _{i = 1}^{N} B_{i} ( \nabla u ) - \, \epsilon \, \hat{a} \, \sum _{i = 1}^{N} B_{i} ( \nabla u ) \, - \, \epsilon \, \varphi ( x^{*} ) \\&\qquad - \, c_{2} \, \sum _{i = 1}^{N} B_{i} ( \nabla u^{m} ) \, - \, 4 h( x^{*} ) \\&\qquad - \, c_{3} \, l( u ) \, \sum _{i = 1} ^{N} B_{i} ( \nabla u ) - c_{4} \, l( u^{m} ) \, \sum _{i = 1}^{N} B_{i} ( \nabla u^{m} ). \end{aligned}$$

So

$$\begin{aligned} A^{j} ( x^{*} )&\ge \big [ \bar{a} \, - \, c_{1} ( \epsilon ) \, - \, \epsilon \, \hat{a} \, \\&\quad - \, c_{3}\, l( u ) \, \big ] \, \sum _{i = 1}^{N} B_{i} ( \nabla u ) + \, \big [ \bar{a} \, - \, \epsilon \, \hat{a} \, c_{2} \, \\&\quad - \, c_{4}\, l( u^{m} ) \, \big ] \, \sum _{i = 1}^{N} B_{i} ( \nabla u^{m} ) \, - \, c_{5} ( \epsilon ). \end{aligned}$$

So we deduce that the sequence $A^{m} (x^{*}) \,$ is not bounded, which is absurd as far as what is in (58).

As a consequence, the sequences $\, u^{m} _{x_{i}} , \, i = 1, \ldots , N, \,\, m \rightarrow \infty$ are bounded.

Let $u^{*} = (u^{*}_{1}, u^{*}_{2}, \ldots , u^{*}_{N}) \, \,$ the limits of subsequence $u^{m} = ( u^{m}_{1}, \ldots , u^{m}_{N}) \, \,$ with $m \, \rightarrow \, \infty .$ Then, taking into account (27), we obtain

$$\begin{aligned} u^{m}_{x_{i}} \, \longrightarrow \, u^{*}_{x_{i}} \quad , \,\,\, i = 1, \ldots , N . \end{aligned}$$

(60)

As a result, from (58), (60) and the fact that $a_{i}^{m} (x^{*} , u, \nabla u)$ are continuous in u (because they are Carathéodory functions), we have

$$\begin{aligned} \sum _{i = 1}^{N}\big ( a_{i}^{m} ( x^{*} , u^{m}, \nabla u^{m} ) - a_{i}^{m} ( x^{*}, u , \nabla u ) \big ) \cdot ( u^{m}_{x_{i}} - u_{x_{i}} ) = 0 , \end{aligned}$$

and from (21) we have, $u^{*}_{x_{i}} = u_{x_{i}}.$ This contradicts the fact that there is no convergence at the point $x^{*}.$

And referring to (27), (60) and the fact that $\, a_{i}^{m} ( x^{*}, u , \nabla u ) \,$ are continuous u, so for $m \rightarrow \infty$ we get

$$\begin{aligned} a_{i}^{m} ( x, u^{m}, \nabla u^{m} )\, \longrightarrow \, a_{i}^{m} ( x, u , \nabla u ) , \,\, i = 1, \ldots , N \,\, \text{ almost } \text{ everywhere } \text{ in } \,\, \Omega . \end{aligned}$$

Using Lemma 3.5 we find the weak convergences

$$\begin{aligned} a_{i}^{m} ( x, u^{m}, \nabla u^{m} ) \rightharpoonup a_{i}^{m} ( x, u , \nabla u ) \,\, \text{ in } \,\, L_{\bar{B}_{i} ( \Omega )},\, i = 1, \ldots , N. \end{aligned}$$

(61)

The weak convergence (48) follows from (61).

Furthermore,to complete the proof, we note that (49) is implied from (46) and (58):

$$\begin{aligned}< A( u^{m} ), u^{m} - u>\,= & lt; A( u^{m} ) - A( u ), u^{m} - u>\\&\quad +\, < A( u ), u^{m} - u > \,\rightarrow 0, \, m \rightarrow \infty . \end{aligned}$$

We’re ending this section by a suitable example, that checks all the above conditions and propositions,

Example 5.1

Let $\Omega$ be an unbounded domain of $\mathbb {R}^{N}, \, ( N \ge 2 )$. By Theorems 3.1 and 4.1 it exists a unique entropy solution based on the Definition 1.1 of the following anisotropic problem $( \mathcal {P}_{1} )$:

$$\begin{aligned} ( \mathcal {P}_{1} ) {\left\{ \begin{array}{ll} \, \tilde{a} \, \displaystyle \sum _{i = 1}^{N} \bar{B}_{i}^{-1} B_{i} ( |\, \nabla u \,| ) + l( u ) \cdot \sum _{i = 1}^{N} B_{i} ( |\, \nabla u \, | ) = f( x ) &\quad \text {in} \,\,\Omega , \\ \, u = 0 &\quad \text {on} \, \partial \Omega . \end{array}\right. } \end{aligned}$$

with $\tilde{a}$ is a positive constant, $l : \mathbb {R} \longrightarrow \mathbb {R}^{+}$ a positive continuous functions such as $l \in L^{1} ( \mathbb {R} ) \cap L^{\infty } ( \mathbb {R} ),$ $f \in L^{1} ( \Omega )$ and

$$\begin{aligned} B( z ) \, = \, | \, z \, |^{b} \, ( \, | \, ln | z | \, | \, + \, 1 \, ), \, \, b \, > \, 1 \end{aligned}$$

satisfying the $\Delta _{2}$-condition.