Existence of renomalized solution for nonlinear elliptic boundary value problem without $\Delta _{2}$ -condition

Abstract

In this paper we will prove in Musielak–Orlicz spaces, the existence of renomalized solution for nonlinear elliptic equations of Leray-Lions type, in the case where the Musielak–Orlicz function $\varphi$ doesn’t satisfy the $\Delta _{2}$ condition while the right hand side f belongs to $W^{-1} E_{\psi }(\Omega )$.

1 Introduction and basic assumptions

This work deals with existence of solutions for strongly nonlinear boundary value problem whose model is:

$$\begin{aligned} \left\{ \begin{array}{c} A(u)-{\text {div}}\Phi (u)+g(x, u, \nabla u)=f \quad in \quad \Omega \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad u \equiv 0, \quad on \quad \partial \Omega \end{array}\right. \end{aligned}$$

(1.1)

where $\Omega$ be a bounded domain of ${\mathbb {R}}^{N}, N \ge 2 ,$ $A(u)=-{\text {div}} a(x, u, \nabla u)$ be a Leray-Lions operator defined from the space $W_{0}^{1} L_{\varphi }(\Omega )$ into its dual $W^{-1} L_{\psi }(\Omega ),$ and $\Phi \in \mathrm {C}^{0}\left( \mathrm {R}, \mathrm {R}^{N}\right)$ . where a is a function satisfying the following conditions :

$$\begin{aligned} a(x, s, \xi ): \Omega \times {\mathbb {R}} \times {\mathbb {R}}^{N} \longrightarrow {\mathbb {R}}^{N} \text {is a Carath}\acute{\mathrm{e}}\text {odory function}. \end{aligned}$$

(1.2)

There exist two Musielak–Orlicz functions $\varphi$ and $\gamma$ such that $\gamma \prec \prec \varphi ,$ a positive function $d(\cdot ) \in E_{\psi }(\Omega )$ and positive constants $k_{1},k_{2}\text { and } k_{3}$ such that for a.e. $x \in \Omega$ and for all $s \in {\mathbb {R}}, \xi \in {\mathbb {R}}^{N}$

$$\begin{aligned}&|a(x, s, \xi )| \le k_{1}\left( d(x)+\psi _{x}^{-1} \gamma (x, k_{2}|s|)\right) +\psi _{x}^{-1} \varphi (x, k_{3}|\xi |); \end{aligned}$$

(1.3)

$$\begin{aligned}&\left( a(x, s, \xi )-a\left( x, s, \xi ^{\prime }\right) \right) \left( \xi -\xi ^{\prime }\right) >0; \end{aligned}$$

(1.4)

$$\begin{aligned}&a(x, s, \xi ) . \xi \ge \alpha \varphi (x,|\xi |). \end{aligned}$$

(1.5)

Furthermore, let $g(x, s, \xi ): \Omega \times {\mathbb {R}} \times {\mathbb {R}}^{N} \longrightarrow {\mathbb {R}}$ be a Carathéodory function such that for a.e. $x \in \Omega$ and for all $s \in {\mathbb {R}}, \xi \in {\mathbb {R}}^{N},$ satisfying the following conditions

$$\begin{aligned}&|g(x, s, \xi )| \le c(x)+b(|s|) \varphi (x,|\xi |); \end{aligned}$$

(1.6)

$$\begin{aligned}&g(x, s, \xi ) s \ge 0; \end{aligned}$$

(1.7)

where $b: {\mathbb {R}}^{+} \longrightarrow {\mathbb {R}}^{+}$ is a continuous positive function which belongs to $L^{1}\left( {\mathbb {R}}^{+}\right)$ and $c(\cdot ) \in L^{1}(\Omega )$ The right-hand side of (1.1) and $\Phi : {\mathbb {R}} \rightarrow {\mathbb {R}}^{N}$ are assumed to satisfy

$$\begin{aligned}&f \in W^{-1} E_{\psi }(\Omega ); \end{aligned}$$

(1.8)

$$\begin{aligned}&\Phi \in {\mathcal {C}}^{0}\left( {\mathbb {R}}, {\mathbb {R}}^{N}\right) . \end{aligned}$$

(1.9)

Note that no growth hypothesis is assumed on the function $\Phi$, which implies that the term $-\text{ div } \Phi (u)$ may be meaningless, even as a distribution.

Several researches deals with the existence solutions of elliptic and parabolic problems under various assumptions and in different contexts (see [1,2,3,4,5,6,7,8,9,10, 13,14,15,16,17,18,19,20, 24,25,28, 35, 37, 39, 40] for more details), indeed we can’t recite all examples; we will just choose some of them, So we mention that:

the problem (1.1) was treated by Boccardo (see [23]) in the case $g\equiv 0$ and for p such that $2-1 / N<p<N$ where he proved the existence and regularity of an entropy solution u that is $u \in W_{0}^{1, q}(\Omega ), \quad q<{\tilde{p}}=\frac{(p-1) N}{N-1},$ $T_{k}(u) \in W_{0}^{1, p}(\Omega ), \quad \forall k>0.$ The same problem have been studied by Diperna and lions in [26] where they introduced the idea of renormalized solutions.

In the framework of variable exponent Sobolev spaces in [12] have proved the existence result of solutions for the problem 1.1 without sign condition involving nonstandard growth.

In the setting of Musielak spaces and in variational case, the existence of a weak solution for the problem (1.1) was treated by Ahmed Oubeid, Benkirane and Sidi El Vally in [11] where ${\text {div}}\Phi \equiv 0.$

Our purpose in this paper is to show the existence of renormalized solutions for problem (1.1) in Musielak Orlicz spaces in the case where the Musielak–Orlicz function $\varphi$ doesn’t satisfy the $\Delta _{2}$ condition,while the right-hand side belongs to $W^{-1} E_{\psi }(\Omega )$, $\Phi \in {\mathcal {C}}^{0}\left( {\mathbb {R}}, {\mathbb {R}}^{N}\right) .$ and a nonlinearity $g(x, s, \xi )$ having natural growth with respect to the gradient.

The paper is organized as follows: In Sect. 2 , we give some preliminaries and background. Section 3 is devoted to some technical lemmas which can be used to our result. In the final Sect. 4, we state our main result and give the prove of an existence solution.

2 Some preliminaries and background

Here we give some definitions and properties that concern Musielak–Orlicz spaces (see [34]).

Let $\Omega$ be an open subset of ${\mathbb {R}}^{n}$, a Musielak–Orlicz function $\varphi$ is a real-valued function defined in $\Omega \times {\mathbb {R}}_{+}$ such that

• (a) $\varphi (x, t)$ is an N-function i.e. convex, nondecreasing, continuous, $\varphi (x, 0)=0,$ $\varphi (x, t)>0$ for all $t>0$ and

  $$\begin{aligned} \begin{aligned} \lim _{t \rightarrow 0} \sup _{x \in \Omega } \frac{\varphi (x, t)}{t}&=0 ,\qquad \lim _{t \rightarrow \infty } \inf _{x \in \Omega } \frac{\varphi (x, t)}{t}&=0 \end{aligned} \end{aligned}$$

• (b) $\varphi (x, t)$ is a measurable function for all $t \ge 0$ .

  Now, let $\varphi _{x}(t)=\varphi (x, t)$ and let $\varphi _{x}^{-1}$ be the non-negative reciprocal function with respect to t,  i.e the function that satisfies

  $$\begin{aligned} \varphi _{x}^{-1}(\varphi (x, t))=\varphi \left( x, \varphi _{x}^{-1}(t)\right) =t \end{aligned}$$

The Musielak–Orlicz function $\varphi$ is said to satisfy the $\Delta _{2}$ -condition if for some $k>0,$ and a non negative function h,  integrable in $\Omega ,$ we have

$$\begin{aligned} \varphi (x, 2 t) \le k \varphi (x, t)+h(x) \text{ for } \text{ all } x \in \Omega \text{ and } t \ge 0. \end{aligned}$$

(2.1)

When (2.1) holds only for $t \ge t_{0}>0,$ then $\varphi$ is said to satisfy the $\Delta _{2}$ -condition near infinity. Let $\varphi$ and $\gamma$ be two Musielak–Orlicz functions, we say that $\varphi$ dominate $\gamma$ and we write $\gamma \prec \varphi ,$ near infinity (resp. globally) if there exist two positive constants c and $t_{0}$ such that for almost all $x \in \Omega$

$\gamma (x, t) \le \varphi (x, c t)$ for all $t \ge t_{0}, \quad \left( \text{ resp. } \text{ for } \text{ all } t \ge 0 \text{ i.e. } t_{0}=0\right)$ We say that $\gamma$ grows essentially less rapidly than $\varphi$ at 0 (resp. near infinity) and we write $\gamma \prec \prec \varphi$ if for every positive constant c we have

$$\begin{aligned} \lim _{t \rightarrow 0}\left( \sup _{x \in \Omega } \frac{\gamma (x, c t)}{\varphi (x, t)}\right) =0, \quad \left( \text{ resp. } \lim _{t \rightarrow \infty }\left( \sup _{x \in \Omega } \frac{\gamma (x, c t)}{\varphi (x, t)}\right) =0\right) \end{aligned}$$

Remark 2.1

(see [29]) If $\gamma \prec \varphi$ near infinity such that $\gamma$ is locally integrable on $\Omega ,$ then $\forall c>0$ there exists a nonnegative integrable function h such that

$$\begin{aligned} \gamma (x, t) \le \varphi (x, c t)+h(x), \text{ for } \text{ all } t \ge 0 \text{ and } \text{ for } \text{ a. } \text{ e. } x \in \Omega . \end{aligned}$$

For a Musielak–Orlicz function $\varphi$ and a measurable function $u: \Omega \longrightarrow {\mathbb {R}},$ we define the functional

$$\begin{aligned} \rho _{\varphi , \Omega }(u)=\int _{\Omega } \varphi (x,|u(x)|) d x \end{aligned}$$

The set $K_{\varphi }(\Omega )=\left\{ u: \Omega \longrightarrow {\mathbb {R}} \text{ measurable } / \rho _{\varphi , \Omega }(u)<\infty \right\}$ is called the Musielak–Orlicz class (or generalized Orlicz class). The Musielak–Orlicz space (the generalized Orlicz spaces) $L_{\varphi }(\Omega )$ is the vector space generated by $K_{\varphi }(\Omega ),$ that is, $L_{\varphi }(\Omega )$ is the smallest linear space containing the set $K_{\varphi }(\Omega ) .$ Equivalently

$$\begin{aligned} L_{\varphi }(\Omega )=\left\{ u: \Omega \longrightarrow {\mathbb {R}} \text{ measurable } / \rho _{\varphi , \Omega }\left( \frac{u}{\lambda }\right) <\infty , \text{ for } \text{ some } \lambda >0\right\} \end{aligned}$$

For a Musielak–Orlicz function $\varphi$ we put: $\psi (x, s)=\sup _{t>0}\{s t-\varphi (x, t)\}, \psi$ is the Musielak–Orlicz function complementary to $\varphi$ (or conjugate of $\varphi$ ) in the sens of Young with respect to the variable s In the space $L_{\varphi }(\Omega )$ we define the following two norms:

$$\begin{aligned} \Vert u\Vert _{\varphi , \Omega }=\inf \left\{ \lambda >0 / \int _{\Omega } \varphi \left( x, \frac{|u(x)|}{\lambda }\right) d x \le 1\right\} \end{aligned}$$

which is called the Luxemburg norm and the so-called Orlicz norm by:

$$\begin{aligned} \Vert |u|\Vert _{\varphi , \Omega }=\sup _{\Vert v\Vert _{\psi } \le 1} \int _{\Omega }|u(x) v(x)| d x \end{aligned}$$

where $\psi$ is the Musielak Orlicz function complementary to $\varphi .$ These two norms are equivalent (see [34])

The closure in $L_{\varphi }(\Omega )$ of the bounded measurable functions with compact support in ${\bar{\Omega }}$ is denoted by $E_{\varphi }(\Omega )$, It is a separable space (see [34], Theorem 7.10) .

We say that sequence of functions $u_{n} \in L_{\varphi }(\Omega )$ is modular convergent to $u \in$ $L_{\varphi }(\Omega )$ if there exists a constant $\lambda >0$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \rho _{\varphi , \Omega }\left( \frac{u_{n}-u}{\lambda }\right) =0. \end{aligned}$$

For any fixed nonnegative integer m we define

$$\begin{aligned} W^{m} L_{\varphi }(\Omega )=\left\{ u \in L_{\varphi }(\Omega ): \forall |\alpha | \le m, D^{\alpha } u \in L_{\varphi }(\Omega )\right\} \end{aligned}$$

and

$$\begin{aligned} W^{m} E_{\varphi }(\Omega )=\left\{ u \in E_{\varphi }(\Omega ): \forall |\alpha | \le m, D^{\alpha } u \in E_{\varphi }(\Omega )\right\} \end{aligned}$$

where $\alpha =\left( \alpha _{1}, \ldots , \alpha _{n}\right)$ with nonnegative integers $\alpha _{i},|\alpha |=\left| \alpha _{1}\right| +\ldots +\left| \alpha _{n}\right|$ and $D^{\alpha } u$ denote the distributional derivatives. The space $W^{m} L_{\varphi }(\Omega )$ is called the Musielak Orlicz Sobolev space.

Let

$$\begin{aligned} {\bar{\rho }}_{\varphi , \Omega }(u)=\sum _{|\alpha | \le m} \rho _{\varphi , \Omega }\left( D^{\alpha } u\right) \text{ and } \Vert u\Vert _{\varphi , \Omega }^{m}=\inf \left\{ \lambda >0: {\bar{\rho }}_{\varphi , \Omega }\left( \frac{u}{\lambda }\right) \le 1\right\} \end{aligned}$$

for $u \in W^{m} L_{\varphi }(\Omega ) .$ These functionals are a convex modular and a norm on $W^{m} L_{\varphi }(\Omega ),$ respectively, and the pair $\left( W^{m} L_{\varphi }(\Omega ),\Vert \Vert _{\varphi , \Omega }^{m}\right)$ is a Banach space if $\varphi$ satisfies the following condition (see [34]):

$$\begin{aligned} \text{ there } \text{ exist } \text{ a } \text{ constant } c_{0}>0 \text{ such } \text{ that } \inf _{x \in \Omega } \varphi (x, 1) \ge c_{0} \end{aligned}$$

(2.2)

The space $W^{m} L_{\varphi }(\Omega )$ will always be identified to a subspace of the product $\prod _{|\alpha | \le m} L_{\varphi }(\Omega )=\Pi L_{\varphi },$ this subspace is $\sigma \left( \Pi L_{\varphi }, \Pi E_{\psi }\right)$ closed.

The space $W_{0}^{m} L_{\varphi }(\Omega )$ is defined as the $\sigma \left( \Pi L_{\varphi }, \Pi E_{\psi }\right)$ closure of ${\mathcal {D}}(\Omega )$ in $W^{m} L_{\varphi }(\Omega ) .$ and the space $W_{0}^{m} E_{\varphi }(\Omega )$ as the (norm) closure of the Schwartz space ${\mathcal {D}}(\Omega )$ in $W^{m} L_{\varphi }(\Omega ).$

Let $W_{0}^{m} L_{\varphi }(\Omega )$ be the $\sigma \left( \Pi L_{\varphi }, \Pi E_{\psi }\right)$ closure of ${\mathcal {D}}(\Omega )$ in $W^{m} L_{\varphi }(\Omega )$ The following spaces of distributions will also be used:

$$\begin{aligned} W^{-m} L_{\psi }(\Omega )=\left\{ f \in D^{\prime }(\Omega ) ; f=\sum _{|\alpha | \le m}(-1)^{|\alpha |} D^{\alpha } f_{\alpha } \text{ with } f_{\alpha } \in L_{\psi }(\Omega )\right\} \end{aligned}$$

and

$$\begin{aligned} W^{-m} E_{\psi }(\Omega )=\left\{ f \in D^{\prime }(\Omega ) ; f=\sum _{|\alpha | \le m}(-1)^{|\alpha |} D^{\alpha } f_{\alpha } \text{ with } f_{\alpha } \in E_{\psi }(\Omega )\right\} \end{aligned}$$

We say that a sequence of functions $u_{n} \in W^{m} L_{\varphi }(\Omega )$ is modular convergent to $u \in W^{m} L_{\varphi }(\Omega )$ if there exists a constant $k>0$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } {\bar{\rho }}_{\varphi , \Omega }\left( \frac{u_{n}-u}{k}\right) =0 \end{aligned}$$

For $\varphi$ and her complementary function $\psi ,$ the following inequality is called the Young inequality (see [34]):

$$\begin{aligned} t s \le \varphi (x, t)+\psi (x, s), \quad \forall t, s \ge 0, x \in \Omega \end{aligned}$$

(2.3)

This inequality implies that

$$\begin{aligned} \Vert u\Vert _{\varphi , \Omega } \le \rho _{\varphi , \Omega }(u)+1 \end{aligned}$$

(2.4)

In $L_{\varphi }(\Omega )$ we have the relation between the norm and the modular

$$\begin{aligned}&{\Vert u\Vert _{\varphi , \Omega } \le \rho _{\varphi , \Omega }(u) \text{ if } \Vert u\Vert _{\varphi , \Omega }>1} \end{aligned}$$

(2.5)

$$\begin{aligned}&{\Vert u\Vert _{\varphi , \Omega } \ge \rho _{\varphi , \Omega }(u) \text{ if } \Vert u\Vert _{\varphi , \Omega } \le 1} \end{aligned}$$

(2.6)

For two complementary Musielak Orlicz functions $\varphi$ and $\psi ,$ let $u \in L_{\varphi }(\Omega )$ and $v \in L_{\psi }(\Omega ),$ then we have the Holder inequality (see [34]):

$$\begin{aligned} \displaystyle {\left| \int _{\Omega } u(x) v(x) d x\right| \le \Vert u\Vert _{\varphi , \Omega }\Vert |v|\Vert _{\psi , \Omega }} \end{aligned}$$

(2.7)

We will use the following technical lemmas.

3 Some technical lemmas

Lemma 3.1

[19] Let $\Omega$ be a bounded Lipschitz domain in ${\mathbb {R}}^{N}$ and let $\varphi$ and $\psi$ be two complementary Musielak–Orlicz functions which satisfy the following conditions:

• (i) There exist a constant $c>0$ such that inf $_{x \in \Omega } \varphi (x, 1) \ge c$.

• (ii) There exist a constant $A>0$ such that for all $x, y \in \Omega$ with $|x-y| \le \frac{1}{2}$ we have

  $$\begin{aligned} \frac{\varphi (x, t)}{\varphi (y, t)} \le t^{\left( \frac{A}{\log \left( \frac{1}{| x-y |}\right) }\right) }, \quad \forall t \ge 1 \end{aligned}$$

  (3.1)
• (iii)
  $$\begin{aligned} \text{ If } D \subset \Omega \text{ is } \text{ a } \text{ bounded } \text{ measurable } \text{ set, } \text{ then } \displaystyle {\int _{D} \varphi (x, 1) d x<\infty } \end{aligned}$$

  (3.2)

• (iv) There exist a constant $C>0$ such that $\psi (x, 1) \le C$ a.e in $\Omega$.

Under this assumptions, ${\mathcal {D}}(\Omega )$ is dense in $L_{\varphi }(\Omega )$ with respect to the modular topology, ${\mathcal {D}}(\Omega )$ is dense in $W_{0}^{1} L_{\varphi }(\Omega )$ for the modular convergence and ${\mathcal {D}}({\bar{\Omega }})$ is dense in $W^{1} L_{\varphi }(\Omega )$ the modular convergence.

Consequently, the action of a distribution S in $W^{-1} L_{\psi }(\Omega )$ on an element u of $W_{0}^{1} L_{\varphi }(\Omega )$ is well defined. It will be denoted by $<S, u>$.

Lemma 3.2

[36] Let $F: {\mathbb {R}} \longrightarrow {\mathbb {R}}$ be uniformly Lipschitzian, with $F(0)=0 .$ Let $\varphi$ be a Musielak–Orlicz function and let $u \in W_{0}^{1} L_{\varphi }(\Omega ) .$ Then $F(u) \in W_{0}^{1} L_{\varphi }(\Omega )$ Moreover, if the set D of discontinuity points of $F^{\prime }$ is finite, we have

$$\begin{aligned} \frac{\partial }{\partial x_{i}} F(u)=\left\{ \begin{array}{cc} {F^{\prime }(u) \frac{\partial u}{\partial x_{i}}\qquad \text{ a.e } i n\{x \in \Omega : u(x) \in D\}} \\ 0 \qquad { \text{ a.e } \text{ in } \{x \in \Omega : u(x) \notin D\}} \end{array}\right. \end{aligned}$$

Lemma 3.3

[29] (Poincare’s inequality) Let $\varphi$ a Musielak Orlicz function which satisfies the assumptions of lemma 3.1, suppose that $\varphi (x, t)$ decreases with respect of one of coordinate of x Then, that exists a constant $c>0$ depends only of $\Omega$ such that

$$\begin{aligned} \int _{\Omega } \varphi (x,|u(x)|) d x \le \int _{\Omega } \varphi (x, c|\nabla u(x)|) d x, \quad \forall u \in W_{0}^{1} L_{\varphi }(\Omega ) \end{aligned}$$

Lemma 3.4

[19] Suppose that $\Omega$ satisfies the segment property and let $u \in$ $W_{0}^{1} L_{\varphi }(\Omega ) .$ Then, there exists a sequence $\left( u_{n}\right) \subset {\mathcal {D}}(\Omega )$ such that

$$\begin{aligned} u_{n} \rightarrow u \text{ for } \text{ modular } \text{ convergence } \text{ in } W_{0}^{1} L_{\varphi }(\Omega ) \end{aligned}$$

Furthermore, if $u \in W_{0}^{1} L_{\varphi }(\Omega ) \cap L^{\infty }(\Omega )$ then $\left\| u_{n}\right\| _{\infty } \le (N+1)\Vert u\Vert _{\infty }$.

Lemma 3.5

Let $\left( f_{n}\right) , f \in L^{1}(\Omega )$ such that

• (i) $f_{n} \ge 0$ a.e in $\Omega$

• (ii) $f_{n} \longrightarrow f$ a.e in $\Omega$

• (iii) $\int _{\Omega } f_{n}(x) d x \longrightarrow \int _{\Omega } f(x) d x$ then $f_{n} \longrightarrow f$ strongly in $L^{1}(\Omega )$

Lemma 3.6

[20] If a sequence $g_{n} \in L_{\varphi }(\Omega )$ converges in measure to a measurable function g and if $g_{n}$ remains bounded in $L_{\varphi }(\Omega ),$ then $g \in L_{\varphi }(\Omega )$ and $g_{n} \rightharpoonup g$ for $\sigma \left( \Pi L_{\varphi }, \Pi E_{\psi }\right)$

Lemma 3.7

(Jensen inequality) [38] Let $\varphi : {\mathbb {R}} \longrightarrow {\mathbb {R}}$ a convex function and g $: \Omega \longrightarrow {\mathbb {R}}$ is function measurable, then

$$\begin{aligned} \varphi \left( \int _{\Omega } g d \mu \right) \le \int _{\Omega } \varphi \circ g d \mu . \end{aligned}$$

Lemma 3.8

(The Nemytskii Operator) [29] Let $\Omega$ be an open subset of ${\mathbb {R}}^{N}$ with finite measure and let $\varphi$ and $\psi$ be two Musielak Orlicz functions. Let $f: \Omega \times {\mathbb {R}}^{p} \longrightarrow {\mathbb {R}}^{q}$ be a Carathodory function such that for a.e. $x \in \Omega$ and all $s \in {\mathbb {R}}^{p}:$

$$\begin{aligned} |f(x, s)| \le c(x)+k_{1} \psi _{x}^{-1} \varphi \left( x, k_{2}|s|\right) \end{aligned}$$

where $k_{1}$ and $k_{2}$ are real positives constants and $c(.) \in E_{\psi }(\Omega )$ Then the Nemytskii Operator $N_{f}$ defined by $N_{f}(u)(x)=f(x, u(x))$ is continuous from

$$\begin{aligned} {\mathcal {P}}\left( E_{M}(\Omega ), \frac{1}{k_{2}}\right) ^{p}=\prod \left\{ u \in L_{M}(\Omega ): d\left( u, E_{M}(\Omega )\right) <\frac{1}{k_{2}}\right\} \end{aligned}$$

into $\left( L_{\psi }(\Omega )\right) ^{q}$ for the modular convergence.

Furthermore if $c(\cdot ) \in E_{\gamma }(\Omega )$ and $\gamma \prec \prec \psi$ then $N_{f}$ is strongly continuous from ${\mathcal {P}}\left( E_{M}(\Omega ), \frac{1}{k_{2}}\right) ^{p}$ to $\left( E_{\gamma }(\Omega )\right) ^{q}$

Lemma 3.9

Let $\Omega$ be a bounded open subset of $R^{N}$ with the segment property. If $u\in (W^{1}_{0}L_{\varphi }(\Omega ))^{N}$ then $\displaystyle { \int _{\Omega } \text{ div } u \ dx = 0}$.

Proof of lemma 3.9

The proof of this lemma is based on [30], Lemma 3.2 ]

4 Main result

We consider the following boundary value problem

$$\begin{aligned} ({\mathcal {P}})\left\{ \begin{array}{c} {A(u)-{\text {div}}\Phi (u)+g(., u, \nabla u)=f \in W^{-1} E_{\psi }(\Omega ) , \quad \text{ in } \Omega } \\ \qquad \qquad \qquad {u \equiv 0, \quad \text{ on } \partial \Omega } \end{array}\right. \end{aligned}$$

Let us define

$$\begin{aligned} {\mathcal {T}}_{0}^{1, \varphi }(\Omega )=\left\{ u \text{ measurable } \quad \text{ such } \text{ that } \quad T_{k}(u) \in W_{0}^{1} L_{\varphi }(\Omega ), \forall k>0\right\} . \end{aligned}$$

As in [21] we define the following notion of renormalized solution, which gives a meaning to a possible solution of $({\mathcal {P}})$

Definition 4.1

Assume that (1.2)–(1.4), (1.6) hold true. A function u is a renormalized solution of the problem $({\mathcal {P}})$ if

$$\begin{aligned} \left\{ \begin{aligned} u \in {\mathcal {T}}_{0}^{1, \varphi }(\Omega ), g(., u, \nabla u) \in L^{1}(\Omega ), g(., u, \nabla u) u \in L^{1}(\Omega )\qquad \qquad \qquad \qquad \qquad \qquad \\ \int _{\Omega } a(x, u, \nabla u) h(u) \nabla v d x+\int _{\Omega } a(x, u, \nabla u) h^{\prime }(u) \nabla u v d x\\ + \int _{\Omega } \Phi (u) h(u)\nabla v d x+\int _{\Omega } \Phi ( u) h^{\prime }(u) \nabla u v d x \\ + \int _{\Omega } g(x, u, \nabla u) h(u) v d x =\int _{\Omega } f h(u) v d x \qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}\right. \end{aligned}$$

(4.1)

for all $h \in W^{1, \infty }({\mathbb {R}})$ such that $h^{\prime }$ has a compact support in ${\mathbb {R}}$, and for all $v \in W_{0}^{1} L_{\varphi }(\Omega ) \cap L^{\infty }(\Omega )$.

The weaker problem (4.1) is obtained by using the test function h(u)v where $h \in$ $W^{1, \infty }({\mathbb {R}}) .$ and $v \in W_{0}^{1} L_{\varphi }(\Omega ) \cap L^{\infty }(\Omega )$ in $({\mathcal {P}})$.

Remark 1

Let us note that in (4.1) every term is meaningful in the distributional sense.

Theorem 4.1

Under assumptions (1.2)–(1.4),(1.6) there exists at least a renormalized solution u in the sense of definition 4.1 of problem $({\mathcal {P}})$.

Let us introduce the truncate operator. For a given constant $k> 0$, we define the function $T_{k}: {\mathbb {R}}\rightarrow {\mathbb {R}}$ as

$$\begin{aligned} T_{k}(s)=\left\{ \begin{array}{rcl} s \text{ if } \left| s\right| \le k,\\ k\frac{s}{\left| s\right| } \text{ if } \left| s\right| > k. \end{array}\right. \end{aligned}$$

4.1 Proof of Theorem 4.1

4.1.1 Approximate problem and a priori estimate

We use an idea contained in [37] (Theorem 1.1), based on the approximation of the original problem and a priori estimate. For $n \in {\mathbb {N}}$, let $\left( f_{n}\right) _{n}$ be a sequence in $W^{-1} E_{\psi }(\Omega ) \cap L^{1}(\Omega )$ such that $f_{n} \longrightarrow f$ in $L^{1}(\Omega )$ with $\left\| f_{n}\right\| _{1} \le \Vert f\Vert _{1}, \phi _{n}(s)=\phi \left( T_{n}(s)\right)$ and $g_{n}(x, s, \xi )=T_{n}(g(x, s, \xi ))$. The following approximate problem

$$\begin{aligned} (P_{n})\left\{ \begin{array}{c} {-{\text {div}}\left( a\left( \cdot , u_{n}, \nabla u_{n}\right) \right) +g_{n}\left( \cdot , u_{n}, \nabla u_{n}\right) =f_{n} +{\text {div}}(\Phi _{n}(u_{n})) \quad \text{ in } D^{\prime }(\Omega )}\\ \\ \qquad \qquad \qquad \qquad \qquad \qquad u_{n} =0 \qquad \qquad \qquad \qquad \qquad \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

has a solution $u_{n}$ in $W^1_0L_\varphi (\Omega )$ .

Now Choosing $u_{n}$ as a function test in problem $(P_{n}),$ we have

$$\begin{aligned} \int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x+\int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla u_{n} d x+\int _{\Omega } g_{n}\left( x, u_{n}, \nabla u_{n}\right) u_{n} d x=\left\langle f, u_{n}\right\rangle \end{aligned}$$

(4.2)

By posing

$$\begin{aligned} {\widetilde{\Phi }}_{n}(t)=\int _{0}^{t} \Phi _{n}(\tau ) d \tau \end{aligned}$$

we obtain

$$\begin{aligned} {\bar{\Phi }}_{n}(0)=0 . \end{aligned}$$

As each component of ${\bar{\Phi }}_{n}$ is uniformly Lipschitizian, and according to [32], Lemma 2], it follows that the function ${\bar{\Phi }}_{n}\left( u_{n}\right)$ belongs to $\left( W_{0}^{1} L_{\varphi }(\Omega )\right) ^{N} .$

therefore by using Lemma 3.9

$$\begin{aligned} \int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla u_{n} d x=\int _{\Omega } {\text {div}}\left( {\widetilde{\Phi }}_{n}\left( u_{n}\right) \right) d x=0 \end{aligned}$$

According to (1.7) and using Young’s inequality, we have

$$\begin{aligned} \left| \int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x \right| \le C_{1}+\frac{\alpha }{2} \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) d x. \end{aligned}$$

(4.3)

which together with (1.5) gives

$$\begin{aligned} \displaystyle { \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) d x\le C_{2}} \end{aligned}$$

(4.4)

Poincare inequality (see Lemma3.3) implies that

$$\begin{aligned} \int _{\Omega } \varphi \left( x, \frac{\left| T_{k}\left( u_{n}\right) \right| }{c}\right) d x \le \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) d x \le c_{2} k \end{aligned}$$

(4.5)

On the other hand we have

$$\begin{aligned} {\qquad \int _{\Omega } g_{n}\left( x, u_{n}, \nabla u_{n}\right) u_{n} d x \le C_{3}} \end{aligned}$$

(4.6)

so it follows that $\left( T_{k}\left( u_{n}\right) \right) _{n}$ and $\left( \nabla T_{k}\left( u_{n}\right) \right) _{n}$ are bounded in $L_{\varphi }(\Omega ),$ Thus

$$\begin{aligned}\left( T_{k}\left( u_{n}\right) \right) _{n} is \text{ bounded } \text{ in } W_{0}^{1} L_{\varphi }(\Omega ),\end{aligned}$$

there exists some $v_{k} \in W_{0}^{1} L_{\varphi }(\Omega )$ such that

$$\begin{aligned} \left\{ \begin{aligned} T_{k}\left( u_{n}\right) \rightharpoonup v_{k} \quad \text{ weakly } \text{ in } W_{0}^{1} L_{\varphi }(\Omega ) \text{ for } \sigma \left( \Pi L_{\varphi }, \Pi E_{\psi }\right) \\ T_{k}\left( u_{n}\right) \longrightarrow v_{k} \qquad \text{ strongly } \text{ in } E_{\psi }(\Omega ). \qquad \qquad \quad \qquad \end{aligned}\right. \end{aligned}$$

(4.7)

Now one suppose that exists a function $\varphi$ satisfies $\lim _{t \rightarrow \infty } \frac{\varphi (t)}{t}=\infty$ and $\varphi (t) \le {\text {ess}} \inf _{x \in \Omega } \varphi (x, t)$ Let $k>0$ large enough, by using (4.5) we have

$$\begin{aligned} \begin{aligned} \varphi (k) {\text {meas}}\left\{ \left| u_{n}\right|>k\right\}&=\int _{\left\{ \left| u_{n}\right|>k\right\} } \varphi \left( \left| T_{k}\left( u_{n}\right) \right| \right) d x \\&\le \int _{\left\{ \left| u_{n}\right| >k\right\} } \varphi \left( x,\left| T_{k}\left( u_{n}\right) \right| \right) d x \le \int _{\Omega } \varphi \left( x,\left| T_{k}\left( u_{n}\right) \right| \right) d x \\&\le c_{3} k \end{aligned} \end{aligned}$$

Hence

$$\begin{aligned} \hbox {meas}\left\{ \left| u_{n}\right| >k\right\} \le \frac{c_{3} k}{\varphi (k)} \longrightarrow 0 \text{ as } k \longrightarrow \infty \end{aligned}$$

For every $\lambda >0,$ we have

$$\begin{aligned} \begin{aligned} \left. \hbox {meas}\left\{ \left| u_{n}-u_{m}\right|>\lambda \right\} \le \lambda \right\}&\le \hbox {meas}\left\{ \left| u_{n}\right|>k\right\} +\hbox {meas}\left\{ \left| u_{m}\right|>k\right\} \\&\quad +\hbox {meas}\left\{ \left| T_{k}\left( u_{n}\right) -T_{k}\left( u_{m} \right) \right| >\lambda \right\} \end{aligned} \end{aligned}$$

(4.8)

then, by using (4.5) one suppose that $\left( T_{k}\left( u_{n}\right) \right) _{n}$ is a Cauchy sequence in measure in $\Omega$, Let $\varepsilon >0,$ then by (4.8) there exists some $k=k(\varepsilon )>0$ such that

$$\begin{aligned} \hbox {meas}\left\{ \left| u_{n}-u_{m}\right| >\lambda \right\} <\varepsilon , \quad \text{ for } \text{ all } n, m \ge h_{0}(k(\varepsilon ), \lambda ) \end{aligned}$$

which means that $\left( u_{n}\right) _{n}$ is a Cauchy sequence in measure in $\Omega ,$ thus converge almost every where to u.

Consequently

$$\begin{aligned} \left\{ \begin{aligned} {u_{n} \rightharpoonup u}&\text{ weakly } \text{ in } W_{0}^{1} L_{\varphi }(\Omega ) \text{ for } \sigma \left( \Pi L_{\varphi }, \Pi E_{\psi }\right) \\ {u_{n} \longrightarrow u}&\text{ strongly } \text{ in } E_{\psi }(\Omega ). \end{aligned}\right. \end{aligned}$$

(4.9)

4.1.2.

In this step we shall show the boundedness of $\left( a\left( \cdot , T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \right) _{n}$ in $\left( L_{\psi }(\Omega )\right) ^{N}$

Let $\vartheta \in E_{\varphi }(\Omega )^{N}$ such that $\Vert \vartheta \Vert _{\varphi , \Omega } \le 1,$ the hypothesis (1.4) gives We have

$$\begin{aligned} \int _{\Omega }\left[ a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \right] \left[ \nabla T_{k}\left( u_{n}\right) -\frac{\vartheta }{k_{3}}\right] d x > 0 \end{aligned}$$

This implies that

$$\begin{aligned}&\int _{\Omega } \frac{1}{k_{3}} a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \vartheta d x \\&\quad \le \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \nabla T_{k}\left( u_{n}\right) d x \\&\qquad -\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \left( \nabla T_{k}\left( u_{n}\right) -\frac{\vartheta }{k_{3}}\right) d x \\&\quad \le c_{2}k-\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \nabla T_{k}\left( u_{n}\right) d x \\&\qquad +\frac{1}{k_{3}} \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \vartheta d x \end{aligned}$$

By using Young’s inequality in the last two terms of the last side and (4.5) we get

$$\begin{aligned}&\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \vartheta d x \le c_{2}k k_{3} \\&\qquad +3 k_{1}\left( 1+k_{3}\right) \int _{\Omega } \psi \left( x, \frac{\left| a\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \right| }{3 k_{1}}\right) d x\\&\qquad +3 k_{1} k_{3} \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) d x+3 k_{1} \int _{\Omega } \varphi (x,|\vartheta |) d x \\&\quad \le c_{2}k k_{3}+3 k_{1} k_{3} c_{2}k +3 k_{1} \\&\qquad +3 k_{1}\left( 1+k_{3}\right) \int _{\Omega } \psi \left( x, \frac{\left| a\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \right| }{3 k_{1}}\right) d x \end{aligned}$$

Now, by using (1.3) and the convexity of $\psi$ we get

$$\begin{aligned} \psi \left( x, \frac{\left| a\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \right| }{3 k_{1}}\right) \le \frac{1}{3}\left( \psi (x, d(x))+\gamma \left( x, k_{2}\left| T_{k}\left( u_{n}\right) \right| \right) +\varphi (x,|\vartheta |)\right) . \end{aligned}$$

Thanks to “Remark 2.1” there exists $h \in L^{1}(\Omega )$ such that

$$\begin{aligned} \gamma \left( x, k_{2}\left| T_{k}\left( u_{n}\right) \right| \right) \le \gamma \left( x, k_{2} k\right) \le \varphi (x, 1)+h(x) ; \end{aligned}$$

then by integrating over $\Omega$ we deduce that

$$\begin{aligned} \begin{array}{l}\displaystyle { \int _{\Omega } \psi \left( x, \frac{\left| a\left( x, T_{k}\left( u_{n}\right) , \frac{v}{k_{3}}\right) \right| }{3 k_{1}}\right) d x \le \frac{1}{3}\left( \int _{\Omega } \psi (x, d(x)) d x+\int _{\Omega } h(x) d x \right. } \\ \displaystyle { \left. \quad +\int _{\Omega } \varphi (x, 1) d x+\int _{\Omega } \varphi (x,|\vartheta |) d x\right) \le c_{k} }, \end{array} \end{aligned}$$

where $c_{k}$ is a constant depending on k. So,

$$\begin{aligned} \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \vartheta d x \le c_{k}, \quad \forall \vartheta \in \left( E_{\varphi }(\Omega )\right) ^{N} \quad \text{ with } \Vert \vartheta \Vert _{\varphi , \Omega }=1 \end{aligned}$$

and thus $\left\| a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \right\| _{\psi , \Omega } \le c_{k},$ which implies that,

$$\begin{aligned} \left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \right) _{n} \text { is bounded in } L_{\psi }(\Omega )^{N}. \end{aligned}$$

(4.10)

4.1.3.

Let us show that :

$$\begin{aligned} \displaystyle {\qquad \lim _{m \rightarrow \infty } \lim _{n \rightarrow \infty } \int _{\left( m \le \left| u_{n}\right| \le m+1 |\right. } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x=0} \end{aligned}$$

Defining

$$\begin{aligned} \theta _{m}(r)=T_{m+1}(r)-T_{m}(r)\qquad \text{ For } \text{ any } m \ge 1, \end{aligned}$$

in view of [32], Lemma2] one get $\theta _{m}\left( u_{n}\right) \in W_{0}^{1} L_{\varphi }(\Omega ).$

Now let us taking $\theta _{m}\left( u_{n}\right)$ as a test function in $(P_{n})$ we have

$$\begin{aligned}&\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \nabla \theta _{m}\left( u_{n}\right) d x+\int _{\Omega } \Phi _{n}\left( u_{n}\right) \nabla \theta _{m}\left( u_{n}\right) d x\\&\quad + \int _{\Omega }g_{n}\left( x, u_{n}, \nabla u_{n}\right) \theta _{m}\left( u_{n}\right) d x =\int _{\Omega } f_{n} \theta _{m}\left( u_{n}\right) d x \end{aligned}$$

Consider,

$$\begin{aligned} \begin{array}{l} \phi (t)=\Phi _{n}(t) \chi _{\{s \in {\mathbb {R}},m\le |s|\le m+1 \}}(t) \\ {\tilde{\phi }}(t)=\displaystyle {\int _{0}^{t} \phi (\tau ) d \tau } \end{array} \end{aligned}$$

hence ${\tilde{\phi }}\left( u_{n}\right) \in \left( W_{0}^{1} L_{\varphi }(\Omega )\right) ^{N}$ (by Lemma3.2). We obtain, by Lemma 3.9,

$$\begin{aligned} \begin{aligned} \int _{\Omega } \Phi _{n}\left( u_{n}\right) \nabla \theta _{m}\left( u_{n}\right) d x&=\int _{\Omega } \Phi _{n}\left( u_{n}\right) \chi _{\{s \in {\mathbb {R}},m\le |s|\le m+1 \}}\left( u_{n}\right) \nabla u_{n} d x \\&=\int _{\Omega } \phi \left( u_{n}\right) \nabla u_{n} d x=\int _{\Omega } {\text {div}}\left( {\tilde{\phi }}\left( u_{n}\right) \right) d x=0 \end{aligned} \end{aligned}$$

Using the sign condition (1.7) we have $g_{n}\left( x, u_{n}, \nabla u_{n}\right) \theta _{m}\left( u_{n}\right) \ge 0$ a.e. in $\Omega ,$ and knowing that $\nabla \theta _{m}\left( u_{n}\right) =\nabla u_{n} \chi _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } \text{ a.e. } \text{ in } \Omega ,$ we get

$$\begin{aligned} \int _{\{m\le |u_{n}|\le m+1 \}} a\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} d x \le \left\langle f, \theta _{m}\left( u_{n}\right) \right\rangle . \end{aligned}$$

It is not difficult to see that

$$\begin{aligned}\left\| \nabla \theta _{m}\left( u_{n}\right) \right\| _{\varphi ,\Omega } \le \left\| \nabla u_{n}\right\| _{\varphi ,\Omega } . \end{aligned}$$

then in view of (4.4) and (4.9) it follows that

$$\begin{aligned} \theta _{m}\left( u_{n}\right) \rightharpoonup \theta _{m}(u) \quad \text{ weakly } \text{ in } W_{0}^{1} L_{\varphi }(\Omega ) \quad \text{ for } \sigma \left( \Pi L_{\varphi }(\Omega ), \Pi E_{\varphi }(\Omega )\right) \end{aligned}$$

Therefore, we get

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\left\{ m \le \left| u_{u}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x \le \left\langle f, \theta _{m}(u)\right\rangle \end{aligned}$$

as $\theta _{m}(u) \rightharpoonup 0$ weakly in $W_{0}^{1} L_{\varphi }(\Omega ,)$ for $\sigma \left( \Pi L_{\varphi }(\Omega ), \Pi E_{\varphi }(\Omega )\right)$ one obtain

$$\begin{aligned} \lim _{m \rightarrow \infty } \lim _{n \rightarrow \infty } \int _{\left( m \le || n_{n}|\le m+1|\right. } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x \le \lim _{m \rightarrow \infty }\left\langle f, \theta _{m}(u)\right\rangle = 0 \end{aligned}$$

By (1.5), we get

$$\begin{aligned} \displaystyle {\qquad \lim _{m \rightarrow \infty } \lim _{n \rightarrow \infty } \int _{\left( m \le \left| u_{k}\right| \le m+1 |\right. } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x=0} \end{aligned}$$

(4.11)

4.1.4.

In this subsubsection we pose $\phi (s)=s e^{\lambda s^{2}}$ where $\lambda =\left( \frac{b(k)}{2 \alpha }\right) ^{2}.$ it is easy to get,

$$\begin{aligned} \text{ for } \text{ all } s \in {\mathbb {R}},\qquad \phi ^{\prime }(s)-\frac{b(k)}{\alpha }|\phi (s)| \ge \frac{1}{2} \end{aligned}$$

(4.12)

For $m \ge k,$ definning

$$\begin{aligned} \psi _{m}(s)=\left\{ \begin{array}{ll} {1} &{} { \text{ if } |s| \le m} \\ {m+1-|s|} &{} { \text{ if } m \le |s| \le m+1} \\ {0} &{} { \text{ if } |s| \ge m+1} \end{array}\right. \end{aligned}$$

Let $\left\{ v_{j}\right\} _{j} \subset D(\Omega )$ be a sequence such that $v_{j} \rightarrow u$ in $W_{0}^{1} L_{\varphi }(\Omega )$ for the modular convergence and a e. in $\Omega$. And let us define the functions

$$\begin{aligned} \theta _{n}^{j}=T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) , \theta ^{j}=T_{k}(u)-T_{k}\left( v_{j}\right) \text{ and } z_{n, m}^{j}=\phi \left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) . \end{aligned}$$

Using $z_{n, m}^{j} \in W_{0}^{1} L_{\varphi }(\Omega )$ as a test function in $(P_{n})$ we get

$$\begin{aligned}&\int _{\Omega }a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla z_{n, m}^{j} d x\nonumber \\&\quad +\int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } \Phi _{n}\left( u_{n}\right) \cdot \nabla u_{n} \psi _{m}^{\prime }\left( u_{n}\right) \phi \left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) d x\nonumber \\&\quad +\int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla \phi \left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \psi _{m}\left( u_{n}\right) d x\nonumber \\&\quad +\int _{\Omega } g_{n}\left( x, u_{n}, \nabla u_{n}\right) z_{n, m}^{j} d x=\int _{\Omega } f z_{n, m}^{j} d x \end{aligned}$$

(4.13)

From now on, we denote by $\epsilon _{i}(n, j), i=0,1,2, \ldots ,$ various sequences of real numbers which tend to zero as n and $j \rightarrow \infty ,$ i.e.,

$$\begin{aligned} \lim _{j \rightarrow +\infty } \lim _{n \rightarrow +\infty } \epsilon _{i}(n, j)=0 \end{aligned}$$

by using (4.7) one has $z_{n, m}^{j} \rightarrow \phi \left( \theta ^{j}\right) \psi _{m}(u)$ weakly in $L^{\infty }(\Omega )$ for $\sigma ^{*}\left( L^{\infty }, L^{1}\right)$ as $n \rightarrow \infty$ which give

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega } f z_{n, m}^{j} d x=\int _{\Omega } f \phi \left( \theta ^{j}\right) \psi _{m}(u) d x \end{aligned}$$

and $\phi \left( \theta ^{j}\right) \rightarrow 0$ weakly in $L^{\infty }(\Omega )$ for $\sigma \left( L^{\infty }, L^{1}\right)$ as $j \rightarrow \infty ,$ we have

$$\begin{aligned} \lim _{j \rightarrow \infty } \int _{\Omega } f \phi \left( \theta ^{j}\right) \psi _{m}(u) d x=0 \end{aligned}$$

Therefore, by denoting

$$\begin{aligned} \int _{\Omega } f z_{n, m}^{j} d x=\epsilon _{0}(n, j), \end{aligned}$$

the divergence lemma implies that

$$\begin{aligned} \int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } \Phi _{n}\left( u_{n}\right) \cdot \nabla u_{n} \psi _{m}^{\prime }\left( u_{n}\right) \phi \left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) d x=0. \end{aligned}$$

The third term in the left-hand side of (4.13) can be written as follows

$$\begin{aligned} \begin{aligned}&\int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla \phi \left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \psi _{m}\left( u_{n}\right) d x \\&\quad =\int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla T_{k}\left( u_{n}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x\\&\qquad -\int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x \end{aligned} \end{aligned}$$

Applying the divergence lemma we have,

$$\begin{aligned} \int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla T_{k}\left( u_{n}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x=0. \end{aligned}$$

By (4.7) one obtain

$$\begin{aligned} \Phi _{n}\left( u_{n}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) \rightarrow \Phi (u) \phi ^{\prime }\left( \theta ^{j}\right) \psi _{m}(u) \text{ a.e. } \text{ in } \Omega \quad as\quad n \rightarrow +\infty \end{aligned}$$

now, we can verify that

$$\begin{aligned} \left\| \Phi _{n}\left( u_{n}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) \right\| _{\varphi ,\Omega } \le \psi \left( x,c_{m} \phi ^{\prime }(2 k)\right) |\Omega |+1 \end{aligned}$$

with $c_{m}=\max _{|t| \le m+1} \Phi (t)$.

Thanks to [33], Theorem 14.6], we have

$$\begin{aligned} \lim _{n \rightarrow +\infty } \int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x=\int _{\Omega } \Phi (u) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta ^{j}\right) \psi _{m}(u) d x \end{aligned}$$

Using the modular convergence of the sequence $\left\{ v_{j}\right\} _{j},$ it follows that

$$\begin{aligned} \lim _{j \rightarrow +\infty } \lim _{n \rightarrow +\infty } \int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x=\int _{\Omega } \Phi (u) \cdot \nabla T_{k}(u) \psi _{m}(u) d x \end{aligned}$$

Then, thanks to Lemma 3.9 we obtain

$$\begin{aligned} \int _{\Omega } \Phi (u) \cdot \nabla T_{k}(u) \psi _{m}(u) d x=0 \end{aligned}$$

Therefore,we denote

$$\begin{aligned} \int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla \phi \left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \psi _{m}\left( u_{n}\right) d x=\epsilon _{1}(n, j). \end{aligned}$$

since $g_{n}\left( x, u_{n}, \nabla u_{n}\right) z_{n m}^{j} \ge 0$ on the set $\left\{ \left| u_{n}\right| >k\right\}$ and $\psi _{m}\left( u_{n}\right) =1$ on the set $\left\{ \left| u_{n}\right| \le k\right\} ,$ by according to 4.13 we get

$$\begin{aligned} \displaystyle { \int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla z_{n m}^{j} d x+\int _{\left\{ \left| u_{n}\right| \le k |\right. } g_{n}\left( x, u_{n}, \nabla u_{n}\right) \phi \left( \theta _{n}^{j}\right) d x \le \epsilon _{2}(n, j) } \end{aligned}$$

(4.14)

For the first term of the left-hand side of (4.14) we can write

$$\begin{aligned}&\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla z_{n, m}^{j} d x =\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) \right. \\&\qquad \left. -\nabla T_{k}\left( v_{j}\right) \right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x\\&\qquad +\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \phi \left( \theta _{n}^{j}\right) \psi _{m}^{\prime }\left( u_{n}\right) d x\\&\quad =\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x\\&\qquad -\int _{|| u_{n}|>k|} a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x\\&\qquad +\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \phi \left( \theta _{n}^{j}\right) \psi _{m}^{\prime }\left( u_{n}\right) d x \end{aligned}$$

therefore

$$\begin{aligned}&\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla z_{n m}^{j} d x\nonumber \\&\quad =\int _{\Omega } \left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \nonumber \\&\qquad \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x\nonumber \\&\qquad +\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{\prime }\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x\nonumber \\&\qquad -\int _{\Omega \backslash \Omega _{j}^{*}} a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x\nonumber \\&\qquad -\int _{\left[ \left| u_{u}\right| >k\right] } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x\nonumber \\&\qquad +\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \phi \left( \theta _{n}^{j}\right) \psi _{m}^{\prime }\left( u_{n}\right) d x \end{aligned}$$

(4.15)

let us define $x_{j}^{s}, s>0,$ and the characteristic function of the subset $\Omega _{j}^{s}=\left\{ x \in \Omega :\left| \nabla T_{k}\left( v_{j}\right) \right| \le s\right\}$.

By fixing m and s,  we will pass to the limit in n and in j in the second, third, fourth and fifth term on the right hand side of (4.15) .

For the second term, we have

$$\begin{aligned}&\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x\\&\quad \rightarrow \int _{\Omega } (a\left( x, T_{k}(u), \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}(u)-\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \phi ^{\prime }\left( \theta ^{j}\right) ) d x \quad \hbox { as }n \rightarrow +\infty \end{aligned}$$

thinks to 3.8, one has

$$\begin{aligned}&a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) x_{j}^{s}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \rightarrow \\&a\left( x, T_{k}(u), \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \phi ^{\prime }\left( \theta ^{j}\right) \text{ strongly } \text{ in } \left( E_{\varphi }(\Omega )\right) ^{N} \text{ as } \text{ n } \rightarrow \infty \end{aligned}$$

and by (4.4)

$$\begin{aligned} \nabla T_{k}\left( u_{n}\right) \rightharpoonup \nabla T_{k}(u) \quad \text{ weakly } \text{ in } \left( L_{\varphi }(\Omega )\right) ^{N} \end{aligned}$$

Let us define $\chi ^{s}$ the characteristic function of the subset

$$\begin{aligned} \Omega ^{s}=\left\{ x \in \Omega :\left| \nabla T_{k}(u)\right| \le s\right\} \end{aligned}$$

As $\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s} \rightarrow \nabla T_{k}(u) \chi ^{s}$ strongly in $\left( E_{\varphi }(\Omega )\right) ^{N}$ as $j \rightarrow \infty ,$ we get

$$\begin{aligned} \int _{\Omega } a\left( x, T_{k}(u), \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}(u)-\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \phi ^{\prime }\left( \theta ^{j}\right) d x \rightarrow 0 \quad \text{ as } j \rightarrow \infty \end{aligned}$$

thus,

$$\begin{aligned} \displaystyle {\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) ) d x=\epsilon _{3}(n, j)} \end{aligned}$$

(4.16)

For third term estimation of (4.15) . It’s it is clear that by (1.5) one can verify that $a(x, s, 0)=0$ for almost every $x \in \Omega \text{ and } \text{ for } \text{ all } s \in {\mathbb {R}}.$

Thus, from (4.10) we have that

$$\begin{aligned} \left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \right) _{n} \text{ is } \text{ bounded } \text{ in } \left( L_{\varphi }(\Omega )\right) ^{N} \text{ for } \text{ all } k \ge 0 . \end{aligned}$$

Therefore, there exist a subsequence still indexed by n and a function $l_{k}$ in $\left( L_{\varphi }(\Omega )\right) ^{N}$ such that

$$\begin{aligned} a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \rightharpoonup l_{k} \text{ weakly } \text{ in } \left( L_{\varphi }(\Omega )\right) ^{N} \text{ for } \sigma \left( \Pi L_{\psi }, \Pi E_{\varphi }\right) . \end{aligned}$$

Then, by using the fact that $\nabla T_{k}\left( v_{j}\right) \chi _{\Omega \backslash \Omega _{j}^{s}} \in \left( E_{\varphi }(\Omega )\right) ^{N},$ we get

$$\begin{aligned}&\int _{\Omega | \Omega _{j}^{s}} a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x\\&\quad \rightarrow \int _{\Omega \backslash \Omega _{j}^{s}} l_{k} \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta ^{j}\right) d x \quad \text{ as } n \rightarrow \infty . \end{aligned}$$

The modular convergence of $\left\{ v_{j}\right\}$ give

$$\begin{aligned} -\int _{\Omega \backslash \Omega _{j}} l_{k} \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta ^{j}\right) d x \rightarrow -\int _{\Omega | \Omega ^{s}} l_{k} \cdot \nabla T_{k}(u) d x \quad \text{ as } j \rightarrow \infty \end{aligned}$$

Consequently

$$\begin{aligned} -\int _{\Omega \backslash \Omega _{j}^{*=s}} a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x=-\int _{\Omega \backslash \Omega ^{s}} l_{k} \cdot \nabla T_{k}(u) d x+\epsilon _{4}(n, j) \end{aligned}$$

(4.17)

For the fourth term, we remark that $\psi _{m}\left( u_{n}\right) =0$ on the subset $\left\{ \left| u_{n}\right| \ge m+1\right\} ,$ then we obtain

$$\begin{aligned}&-\int _{\left\{ \left| u_{n}\right|> k\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x\\&\quad =-\int _{\left\{ \left| u_{n}\right| > k\right\} } a\left( x, T_{m+1}\left( u_{n}\right) , \nabla T_{m+1}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x \end{aligned}$$

By using the same procedure as above we have

$$\begin{aligned}&-\int _{\left\{ \left| u_{n}\right|> k \right\} } a\left( x, T_{m+1}\left( u_{n}\right) , \nabla T_{m+1}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x\\&\quad =-\int _{\{|u|>k\}} l_{m+1} \cdot \nabla T_{k}(u) \psi _{m}(u) d x+\epsilon _{5}(n, j) \end{aligned}$$

By observing that $\nabla T_{k}(u)=0$ on the subset $\{|u|>k\},$ we can write

$$\begin{aligned} -\int _{\left\{ \left| u_{n}\right| >k\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla T_{k}\left( v_{j}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) \psi _{m}\left( u_{n}\right) d x=\epsilon _{5}(n, j) \end{aligned}$$

(4.18)

For the last term of (4.15) we obtain

$$\begin{aligned} \begin{aligned}&\left| \int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \phi \left( \theta _{n}^{j}\right) \psi _{m}^{\prime }\left( u_{n}\right) d x\right| \\&\quad =\left| \int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \phi \left( \theta _{n}^{j}\right) \psi _{m}^{\prime }\left( u_{n}\right) d x\right| \\&\quad \le \phi (2 k) \int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x \end{aligned} \end{aligned}$$

By taking $T_{1}\left( u_{n}-T_{m}\left( u_{n}\right) \right) \in W_{0}^{1} L_{\varphi }(\Omega )$ as test in $(P_{n})$ one has

$$\begin{aligned}&\int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x+\int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } \Phi _{n}\left( u_{n}\right) \cdot \nabla u_{n} d x\\&\quad +\int _{\left\{ \left| u_{x}\right| \ge m\right\} } g_{n}\left( x, u_{n}, \nabla u_{n}\right) T_{1}\left( u_{n}-T_{m}\left( u_{n}\right) \right) d x=\left\langle f, T_{1}\left( u_{n}-T_{m}\left( u_{n}\right) \right) \right\rangle . \end{aligned}$$

by according to Lemma 3.9, we get

$$\begin{aligned} \int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } \Phi _{n}\left( u_{n}\right) \cdot \nabla u_{n} d x=0 \end{aligned}$$

Since $g_{n}\left( x, u_{n}, \nabla u_{n}\right) T_{1}\left( u_{n}-T_{m}\left( u_{n}\right) \right) \ge 0$ on the subset $\left\{ \left| u_{n}\right| \ge m\right\} ,$ we have

$$\begin{aligned} \int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x \le \left\langle f, T_{1}\left( u_{n}-T_{m}\left( u_{n}\right) \right) \right\rangle \end{aligned}$$

By observing f as $f=-{\text {div}} F,$ where $F \in \left( E_{\varphi }(\Omega )\right) ^{N},$ and applying Young’s inequality, we get

$$\begin{aligned} \int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x \le \alpha \quad \int _{\{m \le \left| u_{n}\right| \le m+1\}} \psi \left( x,\frac{2}{\alpha }|F|\right) d x \end{aligned}$$

which implies that

$$\begin{aligned} \left| \int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \phi \left( \theta _{n}^{j}\right) \psi _{m}^{\prime }\left( u_{n}\right) d x\right| \le \alpha \phi (2 k) \quad \int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } \psi \left( x,\frac{2}{\alpha }|F|\right) d x \end{aligned}$$

(4.19)

thinks to (4.15), (4.17), (4.18) and 4.19 we get

$$\begin{aligned}&\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla z_{n, m}^{j} d x \nonumber \\&\quad \ge \int _{\Omega } \left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) x_{j}^{s}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) \right. \nonumber \\&\qquad \left. -\nabla T_{k}\left( v_{j}\right) x_{j}^{s}\right) \phi ^{\prime }\left( \theta _{n}^{j}\right) d x \end{aligned}$$

(4.20)

$$\begin{aligned}&\qquad -\alpha \phi (2 k) \quad \int _{\left\{ m \le \left| u_{k}\right| \le m+1\right\} } \psi \left( x,\frac{2}{\alpha }|F|\right) d x\nonumber \\&\qquad -\int _{\Omega \backslash \Omega ^{s}} l_{k} \cdot \nabla T_{k}(u) d x+e_{6}(n, j) \end{aligned}$$

(4.21)

Now, we turn to the second term on the left-hand side of (4.15) and by using the hypothesis (1.6) one has

$$\begin{aligned} \begin{aligned}&\left| \int _{\left\{ \left| u_{n}\right| \le k\right\} } g_{n}\left( x, u_{n}, \nabla u_{n}\right) \phi \left( \theta _{n}^{j}\right) d x\right| \\&\quad =\left| \int _{\left\{ \left| u_{x}\right| \le k\right\} } g_{n}\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \phi \left( \theta _{n}^{j}\right) d x\right| \\&\quad \le b(k) \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) \left| \phi \left( \theta _{n}^{j}\right) \right| d x+b(k) \int _{\Omega } c(x)\left| \phi \left( \theta _{n}^{j}\right) \right| d x \\&\quad \le \frac{b(k)}{\alpha } \int _{\Omega } a_{n}\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) \left| \phi \left( \theta _{n}^{j}\right) \right| d x+\epsilon _{7}(n, j) \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned}&\left| \int _{\left. \left| u_{k}\right| \le k\right\} } g_{n}\left( x, u_{n}, \nabla u_{n}\right) \phi \left( \theta _{n}^{j}\right) d x\right| \nonumber \\&\quad \le \frac{b(k)}{\alpha } \int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) \right. \nonumber \\&\qquad \left. -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \left| \phi \left( \theta _{n}^{j}\right) \right| d x\nonumber \\&\qquad +\frac{b(k)}{\alpha } \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \left| \phi \left( \theta _{n}^{j}\right) \right| d x\nonumber \\&\qquad +\frac{b(k)}{\alpha } \int _{\Omega } a_{n}\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\left| \phi \left( \theta _{n}^{j}\right) \right| d x+\epsilon _{7}(n, j) \end{aligned}$$

(4.22)

Using the same procedure as above we get

$$\begin{aligned} \frac{b(k)}{\alpha } \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \left| \phi \left( \theta _{n}^{j}\right) \right| d x=\epsilon _{8}(n, j) \end{aligned}$$

and

$$\begin{aligned} \frac{b(k)}{\alpha } \int _{\Omega } a_{n}\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\left| \phi \left( \theta _{n}^{j}\right) \right| d x=\epsilon _{9}(n, j). \end{aligned}$$

thus, we obtain

$$\begin{aligned} \begin{aligned}&\left| \int _{\left\{ \left| u_{n}\right| \le k\right\} } g_{n}\left( x, u_{n}, \nabla u_{n}\right) \phi \left( \theta _{n}^{j}\right) d x\right| \le \frac{b(k)}{\alpha } \int \left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \right. \\&\quad \left. -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \\&\quad \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \left| \phi \left( \theta _{n}^{j}\right) \right| d x+\epsilon _{10}(n, j) \end{aligned} \end{aligned}$$

(4.23)

By combining (4.14),(4.20) and (4.23) we have

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \\&\qquad \times \left( \phi ^{\prime }\left( \theta _{n}^{j}\right) -\frac{b(k)}{\alpha } \left| \phi \left( \theta _{n}^{j}\right) \right| \right) d x\\&\quad \le \int _{\Omega \backslash \Omega ^{s}} l_{k} \cdot \nabla T_{k}(u) d x+\alpha \phi (2 k) \psi \left( x,\frac{2}{\alpha }|F|\right) d x+\epsilon _{11}(n, j) \end{aligned} \end{aligned}$$

thinks to (4.12), we get

$$\begin{aligned}&\int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \nonumber \\&\qquad \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \rho (x) d x\nonumber \\&\quad \le 2 \int _{\Omega \backslash \Omega ^{s}} l_{k} \cdot \nabla T_{k}(u) d x+2 \alpha \phi (2 k) \nonumber \\&\qquad \int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } \psi \left( x,\frac{2}{\alpha }|F|\right) d x+\epsilon _{11}(n, j) \end{aligned}$$

(4.24)

On the other hand, we have

$$\begin{aligned}&\int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi ^{s}\right) d x\\&\quad =\int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \\&\qquad \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) d x\\&\qquad +\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}(u) \chi ^{s}\right) d x\nonumber \\&\qquad -\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi ^{s}\right) d x \nonumber \\&\qquad +\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) d x \end{aligned}$$

We will passe to the limit in n and then in j in the last three terms of the right-hand side of the above equality.

using the same procedure as is done in (4.15) and (4.22), we get

$$\begin{aligned} \begin{aligned}&\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}(u) \chi ^{s}\right) d x=\epsilon _{12}(n, j)\\&\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi ^{s}\right) d x=\epsilon _{13}(n, j)\\&\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) d x=\epsilon _{14}(n, j) \end{aligned} \end{aligned}$$

(4.25)

Therefore,

$$\begin{aligned}&\begin{array}{l} \displaystyle {\int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi ^{s}\right) d x}\end{array}\nonumber \\&\quad =\int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \nonumber \\&\qquad \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) d x+\epsilon _{15}(n, j) \end{aligned}$$

(4.26)

Let $r \le s$. Thinks to (1.4) , (4.24) and (4.26) we have

$$\begin{aligned} \begin{aligned} 0&\le \int _{\Omega ^{r}}\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) d x \\&\le \int _{\Omega ^{s}}\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) d x \\&=\int _{\Omega ^{s}}\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi ^{s}\right) d x \\&\le \int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi ^{s}\right) d x \\&=\int _{\Omega }\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \right) \\&\quad \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) d x+\epsilon _{15}(n, j) \end{aligned} \end{aligned}$$

by passing to the limit in n and then in j one has,

$$\begin{aligned} \begin{aligned} 0&\le \underset{n \rightarrow \infty }{\lim \sup } \int \left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) d x \\&\le 2 \int _{\Omega \backslash \Omega ^{s}} l_{k} \cdot \nabla T_{k}(u) d x+2 \alpha \phi (2 k) \int _{\{m \le |u| \le m+1\}} \psi \left( x,\frac{2}{\alpha }|F|\right) d x.\end{aligned} \end{aligned}$$

Let $s \rightarrow +\infty$ and $m \rightarrow +\infty ,$ using the fact that $l_{k} \cdot \nabla T_{k}(u) \in L^{1}(\Omega ),|F| \in \left( E_{\varphi }(\Omega )\right) ^{N},\left| \Omega \backslash \Omega ^{s}\right| \rightarrow 0$ and $|\{m \le |u| \le m+1\}| \rightarrow 0,$ we obtain

$$\begin{aligned}&\int _{\Omega ^{r}}\left( a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \right) \\&\quad \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) d x \rightarrow 0 \quad \text{ as } n \rightarrow \infty \end{aligned}$$

Thinks to [31] there exists a subsequence of $\left\{ u_{n}\right\}$ still indexed by n such that

$$\begin{aligned} \nabla u_{n} \rightarrow \nabla u \quad \text{ a.e.in } \Omega \end{aligned}$$

(4.27)

Thus, by taking account that (4.7),(4.9) and (4.10) we can apply [33], Theorem 14.6] to obtain $a(x, u, \nabla u) \in \left( L_{\varphi }(\Omega )\right) ^{N}$ and

$$\begin{aligned} a\left( x, u_{n}, \nabla u_{n}\right) \rightharpoonup a(x, u, \nabla u) \text{ weakly } \text{ in } \left( L_{\varphi }(\Omega )\right) ^{N} \text{ for } \sigma \left( \Pi L_{\varphi }(\Omega ), \Pi E_{\psi }(\Omega )\right) . \end{aligned}$$

(4.28)

4.1.5.

Now,we shall prove that

$$\begin{aligned} \text{ for } \text{ every } k>0,\quad T_{k}\left( u_{n}\right) \rightarrow T_{k}(u) \quad \text{ in } W_{0}^{1} L_{\varphi }(\Omega ) \text{ for } \text{ the } \text{ modular } \text{ convegence } \end{aligned}$$

.

From inequality (4.24), we obtain

$$\begin{aligned}&\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) d x \le \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s} d x\\&\quad +\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) x_{j}^{s}\right) \cdot \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) d x \\&\quad +2 \alpha \phi (2 k) \quad \int _{\left\{ m \le | u_{k} | \le m+1\right\} } \psi \left( x,\frac{2}{\alpha }|F|\right) d x\\&\quad +2 \int _{\Omega | \Omega ^{s}} a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) d x+\epsilon _{11}(n, j). \end{aligned}$$

thinks to (4.25), we obtain

$$\begin{aligned}&\int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) d x\\&\quad \le \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s} d x+2 \alpha \phi (2 k)\\&\qquad \int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } \psi \left( x,\frac{2}{\alpha }|F|\right) d x\\&\qquad +2 \int _{\Omega | \Omega ^{s}} a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) d x+\epsilon _{17}(n, j). \end{aligned}$$

the passage to the limitto the limit in n on both sides of this inequality and using (4.28) implies that

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) d x\\&\quad \le \int _{\Omega } a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s} d x+2 \alpha \phi (2 k)\\&\qquad \int _{\{m \le |u| \le m+1\}} \psi \left( x,\frac{2}{\alpha }|F|\right) d x\\&\qquad +2 \int _{\Omega \backslash \Omega ^{s}} a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) d x. \end{aligned}$$

and by passing to the limit in j we obtain

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) d x\\&\quad \le \int _{\Omega } a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) \chi ^{s} d x+ 2 \alpha \phi (2 k)\\&\qquad \int _{\{m \le |u| \le m+1\}} \psi \left( x,\frac{2}{\alpha }|F|\right) d x\\&\qquad +2 \int _{\Omega \backslash \Omega ^{s}} a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) d x. \end{aligned}$$

Let s and $m \rightarrow \infty ,$ we get

$$\begin{aligned} \displaystyle {\limsup _{n \rightarrow \infty } \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) d x \le \int _{\Omega } a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) d x} \end{aligned}$$

Now, thinks to (1.5),(4.4),(4.27) and applying Fatou’s lemma, we get

$$\begin{aligned} \int _{\Omega } a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) d x \le \liminf _{n \rightarrow \infty } \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) d x \end{aligned}$$

thus,

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) d x=\int _{\Omega } a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) d x \end{aligned}$$

In view of Lemma 3.5, we deduce that for every $k>0$

$$\begin{aligned} a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) \rightarrow a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) \quad \text{ strongly } \text{ in } L^{1}(\Omega ) \end{aligned}$$

(4.29)

by vertu of hypothesis (1.5) and using the convexity of $\varphi$ we get

$$\begin{aligned}&\varphi \left( x,\frac{\left| \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right| }{2}\right) \\&\quad \le \frac{1}{2 \alpha } a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \cdot \nabla T_{k}\left( u_{n}\right) +\frac{1}{2 \alpha } a\left( x, T_{k}(u), \nabla T_{k}(u)\right) \cdot \nabla T_{k}(u) \end{aligned}$$

By applying Vitali’s theorem we obtain

$$\begin{aligned} \lim _{|\varepsilon | \rightarrow 0} \sup _{n} \int _{E}\varphi \left( x,\frac{\left| \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right| }{2}\right) d x=0 \end{aligned}$$

Consequently, for every $k>0$

$$\begin{aligned} T_{k}\left( u_{n}\right) \rightarrow T_{k}(u) \quad \text{ in } W_{0}^{1} L_{\varphi }(\Omega ). \end{aligned}$$

for the modular convegence.

4.1.6.

We shall show that

$$\begin{aligned} g_{n}\left( x, u_{n}, \nabla u_{n}\right) \rightarrow g(x, u, \nabla u) \quad in \quad L^{1}(\Omega ) \end{aligned}$$

(4.30)

From (4.9) and (4.27) we have

$$\begin{aligned} g_{n}\left( x, u_{n}, \nabla u_{n}\right) \rightarrow g(x, u, \nabla u) \quad \text{ a.e. } \text{ in } \Omega \end{aligned}$$

(4.31)

Let E be a measurable subset of $\Omega$ and let $m>0 .$ by taking account of (1.5) and (1.6) we obtain

$$\begin{aligned}&\int _{E}\left| g_{n}\left( x, u_{n}, \nabla u_{n}\right) \right| d x=\int _{E \cap \left[ \left| u_{k}\right| \le m\right\} }\left| g_{n}\left( x, u_{n}, \nabla u_{n}\right) \right| d x\\&\qquad +\int _{E \cap \left\{ \left| u_{n}\right| >m\right\} }\left| g_{n}\left( x, u_{n}, \nabla u_{n}\right) \right| d x\\&\quad \le b(m) \int _{E} c(x) d x+b(m) \int _{E} a\left( x, T_{m}\left( u_{n}\right) , \nabla T_{m}\left( u_{n}\right) \right) \cdot \nabla T_{m}\left( u_{n}\right) d x\\&\qquad +\frac{1}{m} \int _{\Omega } g_{n}\left( x, u_{n}, \nabla u_{n}\right) u_{n} d x \end{aligned}$$

By (1.7) and (4.6) it follows that

$$\begin{aligned} \lim _{m \rightarrow \infty } \frac{1}{m} \int _{\Omega } g_{n}\left( x, u_{n}, \nabla u_{n}\right) u_{n} d x=0 \end{aligned}$$

By using (4.29) the sequence

$$\begin{aligned} \left\{ a\left( x, T_{m}\left( u_{n}\right) , \nabla T_{m}\left( u_{n}\right) \right) \cdot \nabla T_{m}\left( u_{n}\right) \right\} _{n} \text{ is } \text{ equi-integrable } , \end{aligned}$$

Consequently

$$\begin{aligned} \lim _{|E| \rightarrow 0} \sup _{n} \int _{E} a\left( x, T_{m}\left( u_{n}\right) , \nabla T_{m}\left( u_{n}\right) \right) \cdot \nabla T_{m}\left( u_{n}\right) d x=0 \end{aligned}$$

This proves that $g_{n}\left( x, u_{n}, \nabla u_{n}\right)$ is equi-integrable.

Therefore, Vitali’s theorem allows us to get

$$\begin{aligned} g(x, u, \nabla u) \in L^{1}(\Omega ), \end{aligned}$$

and

$$\begin{aligned} g_{n}\left( x, u_{n}, \nabla u_{n}\right) \rightarrow g(x, u, \nabla u) \quad \text{ strongly } \text{ in } L^{1}(\Omega ). \end{aligned}$$

(4.32)

4.1.7.

In this subsubsection we prove that

$$\begin{aligned} \lim _{m \rightarrow \infty } \int _{\{m \le |u| \le m+1\}} a(x, u, \nabla u) \cdot \nabla u d x=0 \end{aligned}$$

(4.33)

for any $m \ge 0$ we have

$$\begin{aligned}&\int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x =\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \left( \nabla T_{m+1}\left( u_{n}\right) -\nabla T_{m}\left( u_{n}\right) \right) d x\\&\quad =\int _{\Omega } a\left( x, T_{m+1}\left( u_{n}\right) , \nabla T_{m+1}\left( u_{n}\right) \right) \cdot \nabla T_{m+1}\left( u_{n}\right) d x\\&\qquad -\int _{\Omega } a\left( x, T_{m}\left( u_{n}\right) , \nabla T_{m}\left( u_{n}\right) \right) \cdot \nabla T_{m}\left( u_{n}\right) d x \end{aligned}$$

Thinks to (4.29) and passing to the limit as $n \rightarrow +\infty$ for fixed $m \ge 0$

$$\begin{aligned}&\lim _{n \rightarrow \infty } \int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} d x \\&\quad =\int _{\Omega } a\left( x, T_{m+1}(u), \nabla T_{m+1}(u)\right) \cdot \nabla T_{m+1}(u) d x-\int _{\Omega } a\left( x, T_{m}(u), \nabla T_{m}(u)\right) \nabla T_{m}(u) d x\\&\quad =\int _{\Omega } a(x, u, \nabla u) \cdot \left( \nabla T_{m+1}(u)-\nabla T_{m}(u)\right) d x\\&\quad =\int _{\{m \le |u| \le m+1\}}a(x, u, \nabla u) \cdot \nabla u d x. \end{aligned}$$

according to (4.11), we can pass to the limit as $m \rightarrow +\infty$ in order to have (4.33).

4.1.8.

Finally, in this step thanks to (4.29) and Lemma 3.5, one has

$$\begin{aligned} \displaystyle { a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \rightarrow a(x, u, \nabla u) \cdot \nabla u \quad \text{ strongly } \text{ in } L^{1}(\Omega )} \end{aligned}$$

(4.34)

Let $h \in {\mathbb {C}}_{c}^{1}({\mathbb {R}})$ and $\varrho \in {\mathcal {D}}(\Omega ) .$ we choose $h\left( u_{n}\right) \varrho$ as a test function in $(P_{n})$ we obtain

$$\begin{aligned}&\int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} h^{\prime }\left( u_{n}\right) \varrho d x+\int _{\Omega } \rho (x)a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla \varrho h\left( u_{n}\right) d x\nonumber \\&\quad +\int _{\Omega } \Phi _{n}\left( u_{n}\right) \cdot \nabla \left( h\left( u_{n}\right) \varrho \right) d x+\int _{\Omega } g_{n}\left( x, u_{n}, \nabla u_{n}\right) h\left( u_{n}\right) \varrho d x=\left\langle f, h\left( u_{n}\right) \varrho \right\rangle \end{aligned}$$

(4.35)

Now, we can pass to the limit as $n \rightarrow +\infty$ in each term of equality (4.35). since h and $h^{\prime }$ have a compact support on ${\mathbb {R}},$ there exists a real number $v>0,$ such that supp $h \subset [-v, v]$ and supp $h^{\prime } \subset [-v, v] .$ For $n>v,$ we can have

$$\begin{aligned} \Phi _{n}(t) h(t)=\Phi \left( T_{v}(t)\right) h(t) \quad \text{ and } \quad \Phi _{n}(t) h^{\prime }(t)=\Phi \left( T_{v}(t)\right) h^{\prime }(t) \end{aligned}$$

Moreover,

$$\begin{aligned} \text{ the } \text{ functions } \Phi h \text{ and } \Phi h^{\prime } \text{ belong } \text{ to } \left( {\mathcal {C}}^{0}({\mathbb {R}}) \cap L^{\infty }({\mathbb {R}})\right) ^{N} . \end{aligned}$$

Now we can see that the sequence $\left\{ h\left( u_{n}\right) \varrho \right\} _{n}$ is bounded in $W_{0}^{1} L_{\varphi }(\Omega ).$

Indeed, let $c^{'} >0$ be a positive constant such that $\left\| h\left( u_{n}\right) \nabla \varrho \right\| _{\infty } \le c^{'}$ and $\left\| h^{\prime }\left( u_{n}\right) \varrho \right\| _{\infty } \le c^{'}$.

Thinks to (3.2) we obtain

$$\begin{aligned} \begin{aligned} \int _{\Omega } \varphi \left( x,\frac{\left| \nabla \left( h\left( u_{n}\right) \varrho \right) \right| }{2 c^{'}}\right) d x&\le \int _{\Omega } \varphi \left( x,\frac{\left| h\left( u_{n}\right) \nabla \varrho \right| +\left| h^{\prime }\left( u_{n}\right) \varrho \Vert \nabla u_{n}\right| }{2 c^{'}}\right) d x \\&\le \frac{1}{2} \int _{\Omega } \varphi (x, 1) d x+\frac{1}{2} \int _{\Omega } \varphi \left( x,\left| \nabla T_{M}\left( u_{n}\right) \right| \right) d x \\&\le c \end{aligned} \end{aligned}$$

Which jointly with (4.9) it follows that

$$\begin{aligned} h\left( u_{n}\right) \varrho \rightarrow h(u) \varrho \text{ weakly } \text{ in } W_{0}^{1} L_{\varphi }(\Omega ) \text{ for } \text{ for } \sigma \left( \Pi L_{\varphi }, \Pi E_{\psi }\right) . \end{aligned}$$

(4.36)

Which give

$$\begin{aligned} \left\langle f, h\left( u_{n}\right) \varphi \right\rangle \rightarrow \langle f, h(u) \varphi \rangle . \end{aligned}$$

Let E be a measurable subset of $\Omega .$ we pose $c_{v}=\max _{ |t| \le v} \Phi (t) .$ And denoting by $\Vert v|_{\varphi , \Omega }$ the Orlicz norm of a function $v \in L_{\varphi }(\Omega ) .$ We thinking to the strengthened $H \ddot{o}lder$ inequality with both Orlicz and Luxemburg norms, we have

$$\begin{aligned}&\left\| \Phi \left( T_{v}\left( u_{n}\right) \right) \chi _{E}\right\| _{\psi ,\Omega } =\sup _{\Vert v\Vert _{\varphi ,\Omega } \le 1}\left| \int _{E} \Phi \left( T_{v}\left( u_{n}\right) \right) v d x\right| \\&\quad \le c_{v} \sup _{\Vert v\Vert _{\varphi ,\Omega } \le 1}\left\| \chi _{E}\right\| _{\psi ,\Omega }\Vert v\Vert _{\varphi ,\Omega } \\&\quad \le c_{v}|E| \varphi ^{-1}\left( x,\frac{1}{|E|}\right) \end{aligned}$$

Consequently,

$$\begin{aligned} \lim _{|E| \rightarrow 0} \sup _{n}\left\| \Phi \left( T_{v}\left( u_{n}\right) \right) \chi _{E}\right\| _{(\psi ,\Omega )}=0 \end{aligned}$$

Then, in view of (4.9) and by applying [33], Lemma 11.2], we get

$$\begin{aligned} \Phi \left( T_{v}\left( u_{n}\right) \right) \rightarrow \Phi \left( T_{v}(u)\right) \quad \text{ strongly } \text{ in } \left( E_{\psi }(\Omega )\right) ^{N} \end{aligned}$$

which together with (4.36) allow us to pass to the limit in the third term of (4.35) to obtain

$$\begin{aligned} \int _{\Omega } \Phi \left( T_{v}\left( u_{n}\right) \right) \cdot \nabla \left( h\left( u_{n}\right) \varrho \right) d x \rightarrow \int _{\Omega } \Phi \left( T_{v}(u)\right) \cdot \nabla (h(u) \varrho ) d x \end{aligned}$$

Observing that

$$\begin{aligned} \left| a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} h^{\prime }\left( u_{n}\right) \varrho \right| \le c^{'} a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} \end{aligned}$$

Therefore, thinks to (4.34) and applying Vitali’s theorem, we get

$$\begin{aligned} \int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla u_{n} h^{\prime }\left( u_{n}\right) \varrho d x \rightarrow \int _{\Omega } a(x, u, \nabla u) \cdot \nabla u h^{\prime }(u) \varrho d x \end{aligned}$$

Concerning the second term of (4.35) using the same procedure as above we obtain

$$\begin{aligned} h\left( u_{n}\right) \nabla \varrho \rightarrow h(u) \nabla \varrho \quad \text{ strongly } \text{ in } \left( E_{\varphi }(\Omega )\right) ^{N} \end{aligned}$$

which jointly with (4.28) implies that

$$\begin{aligned} \int _{\Omega } a\left( x, u_{n}, \nabla u_{n}\right) \cdot \nabla \varrho h\left( u_{n}\right) d x \rightarrow \int _{\Omega } a(x, u, \nabla u) \cdot \nabla \varrho h(u) d x \end{aligned}$$

Remark that $h\left( u_{n}\right) \varrho \rightarrow h(u) \varrho$ weakly in $L^{\infty }(\Omega )$ for $\sigma ^{*}\left( L^{\infty }, L^{1}\right)$ with (4.30) we can pass to the limit in the Fourth term of (4.35) in order to have

$$\begin{aligned} \int _{\Omega } g_{n}\left( x, u_{n}, \nabla u_{n}\right) h\left( u_{n}\right) \varrho d x \rightarrow \int _{\Omega } g(x, u, \nabla u) h(u) \varrho d x \end{aligned}$$

Finally, we can pass to the limit in each term of (4.35) so as to obtain

$$\begin{aligned}&\int _{\Omega } a(x, u, \nabla u) \cdot \left[ \nabla \varphi h(u)+h^{\prime }(u) \varrho \nabla u\right] d x+\int \Phi (u) h^{\prime }(u) \varrho \cdot \nabla u d x \\&\quad +\int _{\Omega } \Phi (u) h(u) \cdot \nabla \varrho d x+\int _{\Omega } g(x, u, \nabla u) h(u) \varrho d x =\langle f, h(u) \varrho \rangle \end{aligned}$$

for all $h \in {\mathcal {C}}_{c}^{1}({\mathbb {R}})$ and for all $\varrho \in {\mathcal {D}}(\Omega ) .$ Thus, as well (1.7),(4.6) and (4.31) we apply Fatou’s lemma to get $g(x, u, \nabla u) u \in L^{1}(\Omega )$.

Consequently, thinks to (4.9),(4.28),(4.32),(4.33), the function u is a renormalized solution of problem $({\mathcal {P}})$ .