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.

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.