Existence of solutions for a class of nonlinear elliptic problems with measure data in the setting of Musielak–Orlicz –Sobolev spaces

Abstract

We study the existence of solutions for some nonlinear elliptic problems of the type $-{\text {div}}(b(x, u, \nabla u)+F(x,u))=\nu$ in $\Omega ,$ in the setting of Musielak–Orlicz spaces. The lower order term F verifies the natural growth condition, no $\Delta _{2}$-condition is assumed on the Musielak function, and the datum $\nu$ is assumed to belong to $L^{1}(\Omega )+W^{-1} E_{\psi }(\Omega )$.

1 Introduction and basic assumptions

In this note we will prove an existence of a renormalized solutions for the following nonlinear boundary value problem :

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle B(u)-div\Big (F(x,u)\Big ) =\nu &{}\ \ \text{ in }\ \Omega \\ \displaystyle u =0 &{}\ \ \text{ on }\ {\partial \Omega }, \end{array} \right. \end{aligned}$$

(1.1)

where $\Omega$ is a bounded domain of ${\mathbb {R}}^{N}, N \ge 2,\ B(u)=-{\text {div}}(b(x, u, \nabla u))$ is a Leray-Lions operator defined from the space $W_{0}^{1} L_{\varphi }(\Omega )$ into its dual $W^{-1} L_{\overline{\varphi }}(\Omega ),$ with $\varphi$ and $\overline{\varphi }$ are two complementary Musielak-Orlicz functions and where b is a function satisfying the following conditions:

$$\begin{aligned} b: \Omega \times {\mathbb {R}} \times {\mathbb {R}}^{N} \longrightarrow {\mathbb {R}}^{N} \text{ is } \text{ a } \text{ Carath }\acute{\mathrm{e}}\text {odory function. } \end{aligned}$$

(1.2)

There exist two Musielak-Orlicz functions $\varphi$ and P such that $P \prec \prec \varphi ,$ a positive function $d(x) \in E_{\overline{\varphi }}(\Omega ),\ \alpha >0$ and $k_{i}>0$ for $i=1,\cdots ,4$, such that for a.e. $x \in \Omega$ and all $s \in {\mathbb {R}}$ and all $\xi ,\ \xi ' \in {\mathbb {R}}^{N},\ \xi \not =\xi '$:

$$\begin{aligned} |b(x, s, \xi )|\le & {} k_{1}\Big (d(x)+\overline{\varphi }_{x}^{-1}\left( P\left( x, k_{2}|s|\right) \right) +\overline{\varphi }_{x}^{-1}\left( \varphi \left( x, k_{3}|\xi |\right) \right) \Big ) \end{aligned}$$

(1.3)

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

(1.4)

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

(1.5)

The lower order term F is a Carathéodory function satisfying, for a.e. $x \in \Omega$ and for all $s \in {\mathbb {R}},$ the following condition:

$$\begin{aligned} |F(x, s)| \le c(x) \overline{\varphi }_{x}^{-1} \varphi (x, \alpha _{0}|s|), \end{aligned}$$

(1.6)

where $c(.) \in L^{\infty }(\Omega )$ such that

$$\begin{aligned} \Vert c(.)\Vert _{L^{\infty }} \le \min \left( \frac{\alpha }{\alpha _{0}+1} ; \frac{\alpha }{2\left( \alpha _{0} +1\right) }\right) \text{ and } 0<\alpha _{0}<1. \end{aligned}$$

(1.7)

The right hand side of (1.1) is assumed to satisfy

$$\begin{aligned} \nu \in L^{1}(\Omega )+W^{-1} E_{\overline{\varphi }}(\Omega ):\ \nu =f-{\text {div}}(\phi ) \text{ with } f \in L^{1}(\Omega )\text{ and } \phi \in \left( E_{\overline{\varphi }}(\Omega )\right) ^{N}. \end{aligned}$$

(1.8)

In the usual Sobolev spaces, the concept of renormalized solutions was introduced by Diperna and Lions in [22] for the study of the Boltzmann equations, this notion of solutions was then adapted to the study of the problem (1.1) by Boccardo et al. in [21] when the right hand side is in $W^{-1, p^{\prime }}(\Omega )$ and in the case where the nonlinearity g depends only on x and u,  this work was then studied by Rakotoson in [31] when the right hand side is in $L^{1}(\Omega ),$ and finally by DalMaso et al. in [23] for the case in which the right hand side is general measure data.

On Orlicz-Sobolev spaces and in variational case, Benkirane and Bennouna have studied in [8] the problem (1.1) where $\Phi (x,u)\equiv \Phi (u),$ and the nonlinearity g depends only on x and u under the restriction that the N-function satisfies the $\Delta _{2}$-condition, this work was then extended in [4] by Aharouch, Bennouna and Touzani for N-function not satisfying necessarily the $\Delta _{2}$-condition and $\Phi (x,u)\equiv \Phi (u)$. If g depends also on $\nabla u,$ the problem (1.1) has been solved by Aissaoui Fqayeh, Benkirane, El Moumni and Youssfi in [5] where $\Phi (x,u)\equiv \Phi (u)$, and without assuming the $\Delta _{2}$-condition on the N-function.

In the framework of variable exponent Sobolev spaces, Bendahmane and Wittbold have treated in [7] the nonlinear elliptic equation (1.1) where $a(x,u,\nabla u)=|\nabla u|^{p(x)-2} \nabla u,\ \Phi \equiv 0,\ g\equiv 0$ and where $f\in L^1(\Omega )$, they proved the existence and uniqueness of a renormalized solution in Sobolev space with variable exponents $W_{0}^{1, p(x)}(\Omega ).$

In the variational case of Musielak-Orlicz spaces and in the case where $g \equiv 0$ and $\Phi \equiv 0,$ an existence result for (1.1) has been proved by Benkirane and Sidi El Vally in [10] a when the non-linearity g depends only on x and u. If g depends also on $\nabla u,$ the problem (1.1) has recently been solved by N. El Amarty, B. El Haji and M. El Moumni in [18] where $\Phi (x,u)\equiv \Phi (u).$

and several researches deals with the existence solutions of elliptic and parabolic problems under various assumptions and in different contexts (see [6, 11,12,13,14,15,16, 18,19,20] for more details).

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 Sect. 4, we state our main result and in Sect. 5 we give the proof of an existence solution .

2 Some preliminaries and background

Here we give some definitions and properties that concern Musielak-Orlicz spaces (see [17]). 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) $\varphi (x, .)$ is an N-function for all $x\in \Omega$ (i.e. convex, nondecreasing, continuous, $\varphi (x, 0)=0,\ \varphi (x, t)>0$ for all $t>0$ and $\displaystyle \lim _{t \rightarrow 0} \sup _{x \in \Omega } \frac{\varphi (x, t)}{t}=0$ and $\displaystyle \lim _{t \rightarrow \infty } \inf _{x \in \Omega } \frac{\varphi (x, t)}{t}=\infty$).

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

For a Musielak–Orlicz function $\varphi$, let $\varphi _{x}(t)=\varphi (x, t)$ and let $\varphi _{x}^{-1}$ be the nonnegative reciprocal function with respect to t,  i.e. the function that satisfies

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

The Musielak–Orlicz function $\varphi$ is said to satisfy the $\Delta _{2}$ -condition if for some $k>0,$ and a nonnegative 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 a.e. $x \in \Omega :$

$$\begin{aligned} \gamma (x, t) \le \varphi (x, c t)\ \text {for all}\ t \ge t_{0},\ \text {(resp. for all }~{t \ge 0}~\text { i.e. }~{t_{0}=0)}. \end{aligned}$$

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

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

Remark 1

(see [33]) If $\gamma \prec \prec \varphi$ near infinity, then $\forall \varepsilon >0$ there exists a nonnegative integrable function h,  such that

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

(2.2)

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

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

The set $K_{\varphi }(\Omega )=\Big \{u: \Omega \longrightarrow {\mathbb {R}}\ \text{ measurable/ }\ \rho _{\varphi , \Omega }(u)<\infty \Big \}$ 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:

$$\begin{aligned} \overline{\varphi }(x, s)=\displaystyle \sup _{t>0}\{s t-\varphi (x, t)\}, \end{aligned}$$

Note that $\overline{\varphi }$ is the Musielak-Orlicz function complementary to $\varphi$ (or conjugate of $\varphi$) in the sense of Young with respect to the variable s. In the space $L_{\varphi }(\Omega )$ we define the following two norms:

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

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

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

where $\overline{\varphi }$ is the Musielak-Orlicz function complementary to $\varphi .$ These two norms are equivalent (see [17]). The closure in $L_{\varphi }(\Omega )$ of the bounded measurable functions with compact support in $\overline{\Omega }$ is denoted by $E_{\varphi }(\Omega )$, It is a separable space (see [17], 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

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

and

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

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 for $u \in W^{m} L_{\varphi }(\Omega ):$

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

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 [17]):

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

(2.3)

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

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

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

$$\begin{aligned} W^{-m} L_{\overline{\varphi }}(\Omega )=\Big \{f \in {\mathcal {D}}^{\prime }(\Omega )/\ f=\sum _{|\alpha | \le m}(-1)^{|\alpha |} D^{\alpha } f_{\alpha } \text{ with } f_{\alpha } \in L_{{\varphi }}(\Omega )\Big \} \end{aligned}$$

and

$$\begin{aligned} W^{-m} E_{\overline{\varphi }}(\Omega )=\Big \{f \in {\mathcal {D}}^{\prime }(\Omega )/\ f=\sum _{|\alpha | \le m}(-1)^{|\alpha |} D^{\alpha } f_{\alpha } \text{ with } f_{\alpha } \in E_{{\varphi }}(\Omega )\Big \}. \end{aligned}$$

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

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

We recall that

$$\begin{aligned} \varphi (x,t) \le t \overline{\varphi }^{-1}(\varphi (x,t)) \le 2 \varphi (x,t) \quad \text{ for } \text{ all } t \ge 0. \end{aligned}$$

(2.4)

For $\varphi$ and her complementary function $\overline{\varphi },$ the following inequality is called the Young inequality (see [17]):

$$\begin{aligned} t s \le \varphi (x, t)+\overline{\varphi }(x, s), \quad \forall t, s \ge 0,\ \text {a.e.}\ x \in \Omega . \end{aligned}$$

(2.5)

This inequality implies that

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

(2.6)

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

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

(2.7)

and

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

(2.8)

For two complementary Musielak-Orlicz functions $\varphi$ and $\overline{\varphi },$ let $u \in L_{\varphi }(\Omega )$ and $v \in L_{\overline{\varphi }}(\Omega ),$ then we have the Hölder inequality (see [17]):

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

(2.9)

3 Some technical lemmas

This section concern some technical lemmas that will be used in our main result.

Definition 3.1

We say that a Musielak function $\varphi$ verifies the log-Hölder continuity hypothesis on $\Omega$ if there exists $A>0$ such that

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

$\forall t \ge 1$ and $\forall x, y \in \Omega$ with $|x-y| \le \frac{1}{2}$

Lemma 3.1

[2] Let $\Omega$ be a bounded Lipschitz domain in ${\mathbb {R}}^N(N \ge 2)$ and let $\varphi$ be a Musielak function verifying the log-Hölder continuity such that

$$\begin{aligned} {\bar{\varphi }}(x, 1) \le c_{1} \quad \text { a.e in } \Omega \text { for some } c_{1}>0 \end{aligned}$$

(3.1)

Then ${\mathfrak {D}}(\Omega )$ is dense in $L_{\varphi }(\Omega )$ and in $W_{0}^{1} L_{\varphi }(\Omega )$ for the modular convergence.

Remark 2

Note that if $\lim _{t \rightarrow \infty } \inf _{x \in \Omega } \frac{\varphi (x, t)}{t}=\infty ,$ then (3.1) holds (see [2]).

Example 3.1

Let $p \in P(\Omega )$ a bounded variable exponent on $\Omega ,$ such that there exists a constant $A>0$ such that for all points $x, y \in \Omega$ with $|x-y|<\frac{1}{2},$ we have the inequality

$$\begin{aligned} |p(x)-p(y)| \le \frac{A}{\log \left( \frac{1}{|x-y|}\right) } \end{aligned}$$

We can show that the Musielak function defined by $\varphi (x, t)=t^{p(x)} \log (1+t)$ satisfies the hypothesis of Lemma 3.1.

Proof

(see [2]). $\square$

Lemma 3.2

[2] (Poincare’s inequality: Integral form) Let $\Omega$ be a bounded Lipschitz domain of $R ^{N}(N \ge 2)$ and let $\varphi$ be a Musielak function satisfying the hypothesis of Lemma 3.1. Then there exists $\beta , \eta > 0$ and $\lambda > 0$ depending only on $\Omega$ and $\varphi$ such that

$$\begin{aligned} \int _{\Omega } \varphi (x,|v|) d x \le \beta +\eta \int _{\Omega } \varphi (x, \lambda |\nabla v|) d x \text{ for } \text{ all } v \in W_{0}^{1} L_{\varphi }(\Omega ). \end{aligned}$$

(3.2)

$\square$

Corollary 3.3

[2] (Poincare’s inequality) Let $\Omega$ be a bounded Lipchitz domain of ${\mathbb {R}}^N(N \ge 2)$ and let $\varphi$ be a Musielak function satisfying the same hypothesis of Lemma 3.2. Then there exists $C>0$ such that

$$\begin{aligned} \Vert v\Vert _{\varphi } \le C\Vert \nabla v\Vert _{\varphi } \quad \forall v \in W_{0}^{1} L_{\varphi }(\Omega ). \end{aligned}$$

Lemma 3.4

( [30]) 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 $F(u) \in W_{0}^{1} L_{\varphi }(\Omega )$.

Hawever, if the set D of discontinuity points of $F^{\prime }$ is finite, we obtain

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

Lemma 3.5

[1] (Poincare’s inequality). Let $\varphi$ a Musielak-Orlicz function which satisfies the hypothesis of Lemma 3.1, let $\varphi (x, t)$ decreases with respect of one of coordinate of x, then, that exists $c>0$ depends only of $\Omega$ such that

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

Lemma 3.6

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

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

In addition to this, 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 3.7

Suppose that $\left( g_{n}\right) ,\ g \in L^{1}(\Omega )$ such that

(i) $g_{n} \ge 0$ a.e in $\Omega ,$

(ii) $g_{n} \longrightarrow g$ a.e in $\Omega ,$

(iii) $\displaystyle \int _{\Omega } g_{n}(x) \,dx \longrightarrow \displaystyle \int _{\Omega } g(x) \,dx.$

Then $g_{n} \longrightarrow g$ strongly in $L^{1}(\Omega ).$

Lemma 3.8

[10] If a sequence $h_{n} \in L_{\varphi }(\Omega )$ converges in measure to a measurable function h and if $h_{n}$ remains bounded in $L_{\varphi }(\Omega ),$ then $h \in L_{\varphi }(\Omega )$ and $h_{n} \rightharpoonup h$ for $\sigma \left( \Pi L_{\varphi }, \Pi E_{\overline{\varphi }}\right)$.

Lemma 3.9

[10] Let $v_{n},\ v \in L_{\varphi }(\Omega ).$ If $v_{n} \rightarrow v$ with respect to the modular convergence, then $v_{n} \rightarrow v$ for $\sigma \left( L_{\varphi }(\Omega ), L_{\overline{\varphi }}(\Omega )\right) .$

Lemma 3.10

[25] If $\gamma \prec \varphi$ and $u_{n} \rightarrow u$ for the modular convergence in $L_{\varphi }(\Omega )$ then $u_{n} \rightarrow u$ strongly in $E_{\gamma }(\Omega )$.

Lemma 3.11

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

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

where $k_{1}$ and $k_{2}$ are real positives constants and $c(.) \in E_{\psi }(\Omega )$. Then the Nemytskii Operator $N_{g}$ defined by $N_{g}(u)(x)=g(x, u(x))$ 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. However if $c(\cdot ) \in E_{\gamma }(\Omega )$ and $\gamma \prec \prec \psi$ then $N_{g}$ is strongly continuous from ${\mathcal {P}}\left( E_{M}(\Omega ), \frac{1}{k_{2}}\right) ^{p}$ to $\left( E_{\gamma }(\Omega )\right) ^{q}$.

4 Main result

We now give the definition of a renormalized solution of (1.1).

Definition 4.1

A measurable function $u: \Omega \rightarrow {\mathbb {R}}$ is called a renormalized solution of (1.1) if:

$$\begin{aligned} T_{k}(u) \in W_{0}^{1} L_{\varphi }(\Omega ) \quad \text {and} \quad b(x, u, \nabla u) \in \left( L_{\overline{\varphi }}(\Omega )\right) ^{N}, \end{aligned}$$

(4.1)

$$\begin{aligned} \displaystyle \lim _{m \rightarrow +\infty } \int _{\{x \in \Omega :\ \ m \le |u(x)| \le m+1\}} b(x, u, \nabla u) \nabla u \,dx=0, \end{aligned}$$

(4.2)

and for every function $h \in C_{c}^{1}({\mathbb {R}})$ such that

$$\begin{aligned} -{\text {div}}\Big ( b(x, u, \nabla u) h(u)\Big ) -{\text {div}}\Big (F(x,u) h(u)\Big )+h^{\prime }(u) F(x,u) \nabla u \end{aligned}$$

(4.3)

$$\begin{aligned} =f h(u)-{\text {div}}(\phi h(u))+h^{\prime }(u) \phi \nabla u \quad \text{ in } {\mathcal {D}}^{\prime }(\Omega ). \end{aligned}$$

Remark 3

Every term in equation (4.3) is meaningful in the distributional sense. Indeed, for $h \in C_{c}^{1}({\mathbb {R}})$ and $u\in W_{0}^{1} L_{\varphi }(\Omega ),$ then $h(u)\in W^{1} L_{\varphi }(\Omega )$ and for V in ${\mathcal {D}}(\Omega )$ the function $V h(u)\in W_{0}^{1} L_{\varphi }(\Omega ).$ Since ${\text {div}} \Big (b(x, u, \nabla u)\Big ) \in W^{-1} L_{\overline{\varphi }}(\Omega ),$ we have for every $V \in {\mathcal {D}}(\Omega )$:

$$\begin{aligned} \Big \langle {\text {div}}\Big (b(x, u, \nabla u)\Big )h(u)\ ;\ V\Big \rangle _{{\mathcal {D}}^{\prime }(\Omega ), {\mathcal {D}}(\Omega )} =\Big \langle {\text {div}}\Big (b(x, u, \nabla u)\Big )\ ;\ V h(u)\Big \rangle _{W^{-1} L_{\overline{\varphi }}(\Omega ), W_{0}^{1} L_{\varphi }(\Omega )} \end{aligned}$$

Finally, $F(x,u) h(u)\in \left( L^{\infty }(\Omega )\right) ^{N},\ F(x,u) h^{\prime }(u)\in \left( L^{\infty }(\Omega )\right) ^{N},\ {\text {div}}\Big (F(x,u) h(u)\Big ) \in W^{-1} L_{\overline{\varphi }}(\Omega )$ and $F(x,u) h^{\prime }(u)\nabla u \in L_{\varphi }(\Omega ).$

Our main result is the following

Theorem 4.1

Under assumptions (1.2)-(1.8) there exists at least a renormalized solution of Problem (1.1).

Remark 4

Actually the original equation (1.1) will be recovered whenever $h(u) \equiv 1$ but unfortunately this cannot happen in general strong additional requirements on u. Therefore, (4.3) is to be viewed as a weaker form of (1.1).

Remark 5

Generalized Orlicz spaces (Musielak-Orlicz-sobolev spaces), Orlicz spaces and $L^{p(\cdot )}$-spaces have different nature, and neither of them is a subset of the other.

Let us list some techniques from the classical case which do not work in $L^{p(\cdot )}{ }_{-}$ spaces and some additional ones that do not work in the generalized Orlicz case. Orlicz spaces are similar to $L^p$-spaces in many regards, but some differences exist.

• Exponents cannot be moved outside the $\Phi$-function, i.e. $\varphi \left( t^\gamma \right) \ne \varphi (t)^\gamma$ in general.

• The formula $\varphi ^{-1}\left( \int _{\Omega } \varphi (|f|) d x\right)$ does not define a norm. Techniques which do not work in $L^{p(\cdot )}$-spaces (from [24], pp. 9–10]):

• The space $L^{p(\cdot )}$ is not rearrangement invariant; the translation operator $T_h$ : $L^{p(\cdot )} \rightarrow L^{p(\cdot )}, T_h f(x):=f(x+h)$ is not bounded; Young’s convolution inequality $\Vert f * g\Vert _{p(\cdot )} \leqslant c\Vert f\Vert _1\Vert g\Vert _{p(\cdot )}$ does not hold [24], Section 3.6].

• The formula

  $$\begin{aligned} \int _{\Omega }|f(x)|^p d x=p \int _0^{\infty } t^{p-1}|\{x \in \Omega :|f(x)|>t\}| d t \end{aligned}$$

  has no variable exponent analogue.

• Maximal, Poincaré, Sobolev, etc., inequalities do not hold in a modular form. For instance, A. Lerner showed that the inequality

  $$\begin{aligned} \int _{{\mathbb {R}}^n}|M f|^{p(x)} d x \leqslant c \int _{{\mathbb {R}}^n}|f|^{p(x)} d x \end{aligned}$$

  holds if and only if $p \in (1, \infty )$ is constant [29], Theorem 1.1]. For the Poincaré inequality see [24], Example 8.2.7] and the discussion after it.

• Interpolation is not so useful, since variable exponent spaces never result as an interpolant of constant exponent spaces (see Sect. 5.5).

• Solutions of the $p(\cdot )$-Laplace equation are not scalable, i.e. $\lambda u$ need not be a solution even if u is [24], Example 13.1.9]. New obstructions in generalized Orlicz spaces:

• We cannot estimate $\varphi (x, t) \lesssim \varphi (y, t)^{1+\varepsilon }+1$ even when $|x-y|$ is small, because of lack of polynomial growth. This complicates e.g. the use of higher integrability in PDE proofs.

• It is not always the case that $\chi _E \in L^{\varphi }(\Omega )$ when $|E|<\infty$.

5 Proof of Theorem 4.1

Throughout the paper, $T_{k}$ denotes the truncation function at height $k\ge 0:$

$$\begin{aligned} T_{k}(s)=\max (-k, \min (k, s)) \end{aligned}$$

5.1 Approximate problem

For $n \in {\mathbb {N}}^*,$ let define the following approximations of $f$ and $\Phi .$ Let $f_{n}$ be a sequence of $L^{\infty }(\Omega )$ functions that converge strongly to f in $L^{1}(\Omega ),$ and $\left\| f_{n}\right\| _{L^{1}} \le$ $\Vert f\Vert _{L^{1}}.$ Let $F_{n}\left( x, s\right) =F\left( x, T_{n}\left( s\right) \right)$. Then we consider the approximate Eq. (1.1) for $n\ge 1:\ u_{n} \in W_{0}^{1} L_{\varphi }(\Omega )$

$$\begin{aligned} -{\text {div}}\Big ( b\left( x, u_{n}, \nabla u_{n}\right) \Big )+{\text {div}}\Big ( F_{n}\left( x,u_{n}\right) \Big ) =f_{n}-{\text {div}}(\phi ) \quad {\text {in}} {\mathcal {D}}^{\prime }(\Omega ). \end{aligned}$$

(5.1)

there exists at last one solution $u_{n} \in$ $W_{0}^{1} L_{\varphi }(\Omega )$ of (5.1) (see [26]).

5.2 A priori estimates

Choosing $T_{k}(u_{n})$ as a test function in (5.1), we get

$$\begin{aligned} \begin{array}{clll} &{}\displaystyle \int _{\Omega } b_{n}(x, u_{n}, \nabla u_{n}) \nabla T_{k}\left( u_{n}\right) \,dx +\int _{\Omega } F_{n}(x, u_{n}) \nabla T_{k}\left( u_{n}\right) \,dx\\ &{}\quad \le k \left\| f_{n}\right\| _{L^{1}(\Omega )}+\displaystyle \int _{\Omega } \phi \nabla T_{k}(u_{n}) \,dx. \end{array} \end{aligned}$$

(5.2)

By (1.6), Lemma 3.5 and Young inequality, we obtain:

$$\begin{aligned} \begin{array}{cll} &{}\displaystyle { \int _{\Omega } F_{n}(x, u_{n}) \nabla T_{k}\left( u_{n}\right) \,dx } \\ &{}\quad \displaystyle { \le \Vert c(.)\Vert _{L^{\infty }(\Omega )}\left[ \alpha _{0} \int _{\Omega } \varphi \left( x, u_{n}\right) T_{k}\left( u_{n}\right) \,dx\right. } +\displaystyle \left. \int _{\Omega } \varphi \left( x,\left| \nabla u_{n}\right| \right) T_{k}\left( u_{n}\right) d x\right] . \\ &{}\quad \displaystyle {} \le \Vert c(.)\Vert _{L^{\infty }}\left( \alpha _{0} +1\right) \int _{Q_{\tau }} \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) d x d t.\end{array} \end{aligned}$$

(5.3)

Recall that

$$\begin{aligned} \begin{array}{c}\displaystyle { \int _{\Omega } \phi \nabla T_{k}\left( u_{n}\right) \,dx \le \frac{\alpha }{2} \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}(u_{n})\right| \right) \,dx}+c(\Omega,N,\alpha,\phi) . \end{array} \end{aligned}$$

(5.4)

return to (5.2) and using (5.3) and (5.4) we get

$$\begin{aligned} \begin{array}{clll} &{}\displaystyle {} \int _{\Omega } b_{n}\left( x,u_{n}, \nabla u_{n}\right) \nabla T_{k}\left( u_{n}\right) d x d t \displaystyle {} \le k \left\| f_{n}\right\| _{L^{1}(\Omega )}+\left[ \Vert c(.)\Vert _{L^{\infty }}\left( \alpha _{0} +1\right) + \frac{\alpha }{2} \right] \\ &{}\quad \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) d x d t \end{array} \end{aligned}$$

(5.5)

by using (1.5) we get

$$\begin{aligned} \begin{aligned}&\int _{\Omega } b_{n}\left( x,u_{n}, \nabla u_{n}\right) \nabla T_{k}\left( u_{n}\right) d x d t \le \frac{\left[ \Vert c(.)\Vert _{L^{\infty }}\left( \alpha _{0} +1\right) + \frac{\alpha }{2} \right] }{\alpha }\\&\quad \int _{\Omega } b_{n}\left( x,u_{n}, \nabla u_{n}\right) \nabla T_{k}\left( u_{n}\right) d x d t +k\left\| f_{n}\right\| _{L^{1}\left( \Omega \right) }, \end{aligned} \end{aligned}$$

thus

$$\begin{aligned} \left[ \frac{1}{2}-\frac{\left[ \Vert c(.)\Vert _{L^{\infty }}\left( \alpha _{0} +1\right) \right] }{\alpha }\right] \int _{\Omega } b\left( x, u_{n}, \nabla u_{n}\right) \nabla \left( T_{k}\left( u_{n}\right) \right) \,dx \begin{aligned}&\le k c_{1}, \end{aligned} \end{aligned}$$

We take $\displaystyle \frac{1}{c_{2}}=\left[ \frac{1}{2}-\frac{\left[ \Vert c(.)\Vert _{L^{\infty }}\left( \alpha _{0} +1\right) \right] }{\alpha }\right] .$ Then we deduce that

By (1.7) we have $c_{2}>0$ where $C=c_{1} c_{2} .$ And by (1.5) we have

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

(5.6)

So it follows that $\left( T_{k}\left( u_{n}\right) \right) _{n} is \text{ bounded } \text{ in } W_{0}^{1} L_{\varphi }(\Omega ),$ then there exists some $v_{k} \in W_{0}^{1} L_{\varphi }(\Omega )$ such that

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

(5.7)

On the other hand, using (5.6), we have

$$\begin{aligned}&\displaystyle \inf _{x \in \Omega } \varphi \left( x, \frac{k}{\delta }\right) {\text {meas}}\left\{ \left| u_{n}\right|>k\right\} \le \int _{\left\{ \left| u_{n}\right| >k\right\} } \varphi \left( x, \frac{\left| T_{k}\left( u_{n}\right) \right| }{\delta }\right) \,dx\\&\quad \le \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) \,dx \le k C. \end{aligned}$$

Then

$$\begin{aligned} {\text {meas}}\left\{ \left| u_{n}\right| >k\right\} \le \frac{ k C}{\inf _{x \in \Omega } \varphi \left( x, \frac{k}{\delta }\right) } \end{aligned}$$

for all $n\ge 1$ and for all $k\ge 1$. Assuming that there exists a positive function $\overline{\varphi }$ such that $\displaystyle \lim _{t \rightarrow \infty } \frac{\overline{\varphi }(t)}{t}=+\infty$ and $\overline{\varphi }(t) \le ess\inf _{x \in \Omega } \varphi (x, t),\ \forall t \ge 0.$ Thus, we get

$$\begin{aligned} \lim _{k \rightarrow \infty } {\text {meas}}\left\{ \left| u_{n}\right| >k\right\} =0. \end{aligned}$$

(5.8)

Let $\eta >0$ and $\epsilon >0$ then

$$\begin{aligned} {\text {meas}}\left\{ \left| u_{n}-u_{m}\right|>\eta \right\} \le {\text {meas}}\left\{ \left| u_{n}\right|>k\right\} +{\text {meas}}\left\{ \left| u_{m}\right|>k\right\} +{\text {meas}}\left\{ \left| T_{k}\left( u_{n}\right) -T_{k}\left( u_{m}\right) \right| >\eta \right\} \end{aligned}$$

then, by using (5.7) 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 (5.8) there exists some $k=k(\varepsilon )>0$ such that

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

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

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

(5.9)

5.3 Boundedness of $\left( b\left( x, u_{n}, \nabla u_{n}\right) \right) _{n}$ in $\left( L_{\overline{\varphi }}(\Omega )\right) ^{N}$

Let $\vartheta \in \left( E_{\varphi }(\Omega )\right) ^{N}$ such that

$\Vert \vartheta \Vert _{\varphi , \Omega }=1,$ we have

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

This implies that

$$\begin{aligned}&\displaystyle \int _{\Omega } \frac{1}{k_{3}} b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \vartheta \,dx \nonumber \\&\quad \le \int _{\Omega } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \nabla T_{k}\left( u_{n}\right) \,dx -\int _{\Omega } b\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \left( \nabla T_{k}\left( u_{n}\right) -\frac{\vartheta }{k_{3}}\right) \,dx \nonumber \\&\quad \le k C_{1}+C_{2}-\int _{\Omega } b\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \nabla T_{k}\left( u_{n}\right) \,dx +\frac{1}{k_{3}} \int _{\Omega } b\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \vartheta \,dx. \end{aligned}$$

(5.10)

By using Young’s inequality in the last two terms of the last side and (5.6) we get

$$\begin{aligned}&\displaystyle \int _{\Omega } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \vartheta \,dx \nonumber \\&\quad \displaystyle \le k_{3}(k C_{1}+C_{2}) +3 k_{1}\left( 1+k_{3}\right) \int _{\Omega } \overline{\varphi }\left( x, \frac{\left| b\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \right| }{3 k_{1}}\right) \,dx \nonumber \\&\qquad \displaystyle +3 k_{1} k_{3} \int _{\Omega } \varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) d x+3 k_{1} \int _{\Omega } \varphi (x,|\vartheta |) \,dx \nonumber \\&\quad \le k_{3}(k C_{1}+C_{2}) +3 k_{1} k_{3} (k C_{1}+C_{2}) +3 k_{1} \displaystyle +3 k_{1}\left( 1+k_{3}\right) \int _{\Omega } \overline{\varphi }\left( x, \frac{\left| b\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \right| }{3 k_{1}}\right) \,dx \end{aligned}$$

(5.11)

Now, by using (1.3) and the convexity of $\overline{\varphi }$ we get

$$\begin{aligned} \overline{\varphi }\left( x, \frac{\left| b\left( x, T_{k}\left( u_{n}\right) , \frac{\vartheta }{k_{3}}\right) \right| }{3 k_{1}}\right) \le \frac{1}{3}\Big (\overline{\varphi }(x, d(x))+P\left( x, k_{2}\left| T_{k}\left( u_{n}\right) \right| \right) +\varphi (x,|\vartheta |)\Big ) \end{aligned}$$

(5.12)

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

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

then by integrating over $\Omega$ we deduce that

$$\begin{aligned}&\displaystyle \int _{\Omega } \overline{\varphi }\left( x, \frac{\left| b\left( x, T_{k}\left( u_{n}\right) , \frac{v}{k_{3}}\right) \right| }{3 k_{1}}\right) \,dx \nonumber \\&\quad \displaystyle \le \frac{1}{3}\left( \int _{\Omega } \overline{\varphi }(x, c(x)) \,dx +\int _{\Omega } h(x) \,dx +\int _{\Omega } \varphi (x, 1) \,dx+\int _{\Omega } \varphi (x,|\vartheta |) \,dx\right) \le c_{k}', \end{aligned}$$

(5.13)

where $c_{k}'$ is a constant depending on k,  then $\forall \vartheta \in \left( E_{\varphi }(\Omega )\right) ^{N}$ with $\Vert \vartheta \Vert _{\varphi , \Omega }=1$ we have $\displaystyle \int _{\Omega } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \vartheta d x \le c_{k}',$ and thus $\left\| b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \right\| _{\overline{\varphi }, \Omega } \le c_{k}',$ which implies that

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

(5.14)

5.4 Renormalization identity for the approximate solutions

Consider the function $Z_{m}\left( u_{n}\right) =T_{1}\left( u_{n}-T_{m}\left( u_{n}\right) \right)$ and by taking $Z_{m}\left( u_{n}\right)$ as test function in (5.1) we obtain

$$\begin{aligned}&\displaystyle \int _{\Omega } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \nabla Z_{m}\left( u_{n}\right) \,dx \displaystyle +\int _{\Omega } F_{n}\left( x, u_{n}\right) \nabla Z_{m}\left( u_{n}\right) \,dx \nonumber \\&\quad \displaystyle =\int _{\Omega } f_{n} Z_{m}\left( u_{n}\right) d x+\int _{\Omega } \phi \nabla Z_{m}\left( u_{n}\right) \,dx. \end{aligned}$$

(5.15)

By the same argument used in a priori estimates, we get

$$\begin{aligned}&\displaystyle \int _{\Omega } \varphi \left( x,\left| \nabla Z_{m}\left( u_{n}\right) \right| \right) \,dx \displaystyle \le C\left[ \int _{\Omega } f_{n} Z_{m}\left( u_{n}\right) d x\right. \left. + \int _{\Omega } \overline{\varphi }\left( x, \frac{|\phi |}{\epsilon _{1}}\right) Z_{m}\left( u_{n}\right) d x\right] \nonumber \\&\quad \displaystyle + C \int _{\left\{ m \le u_{n} \le m+1 \right\} } \overline{\varphi }\left( x, \frac{|\phi |}{\epsilon _{1}}\right) \,dx \end{aligned}$$

(5.16)

where $\displaystyle \frac{1}{C}=\left[ \frac{1}{2}-\left( \frac{\Vert c(\cdot )\Vert _{L^{\infty }(\Omega )}+\epsilon _{1}}{\alpha }\right) \right] .$ In order to pass to the limit in (5.16) as $n \rightarrow +\infty ,$ we use the pointwise convergence of $u_{n}$ and strongly convergence in $L^{1}(\Omega )$ of $f_{n},$ we get

$$\begin{aligned}&\displaystyle \lim _{n \rightarrow +\infty } \int _{\Omega } \varphi \left( x,\left| \nabla Z_{m}\left( u_{n}\right) \right| \right) \,dx \le C\left[ \int _{\Omega } f Z_{m}(u) d x\right. \left. + \int _{\Omega } \overline{\varphi }\left( x, \frac{|\phi |}{\epsilon _{1}}\right) Z_{m}(u) d x\right] \nonumber \\&\quad \displaystyle C \int _{\{m \le u \le m+1 \}} \overline{\varphi }\left( x, \frac{|\phi |}{\epsilon _{1}}\right) \,dx \end{aligned}$$

(5.17)

Thanks to Lebesgue’s theorem and passing to the limit as $m \rightarrow +\infty ,$ in every term of the right-hand side of the previous inequalities, we obtain

$$\begin{aligned} \lim _{m \rightarrow +\infty } \lim _{n \rightarrow +\infty } \int _{\Omega } \varphi \Big (x,\left| \nabla Z_{m}\left( u_{n}\right) \right| \Big ) \,dx=0. \end{aligned}$$

(5.18)

Using (1.6) and Young inequality, for $n>m+1$ we have

$$\begin{aligned}&\displaystyle { \int _{\Omega }\left| F_{n}\left( x, u_{n}\right) \nabla Z_{m}\left( u_{n}\right) \right| d x \le \int _{\left\{ m \le u_{n} \le m+1\right\} } \varphi \left( x, \alpha _{0}\left| T_{m+1}\left( u_{n}\right) \right| \right) d x } \nonumber \\&\quad \displaystyle { +\int _{\Omega } \varphi \left( x,\left| \nabla Z_{m}\left( u_{n}\right) \right| \right) d x}. \end{aligned}$$

(5.19)

Thanks to Lebesgue’s theorem, and by the pointwise convergence of $u_{n}$ we can have

$$\begin{aligned}&\displaystyle \lim _{n \rightarrow +\infty } \int _{\Omega }\Big |F_{n}\left( x, u_{n}\right) \nabla Z_{m}\left( u_{n}\right) \Big | \,dx \le \int _{\{ m \le u \le m+1\}} \varphi \left( x, \alpha _{0}\left| T_{m+1}(u)\right| \right) d x\nonumber \\&\quad \displaystyle { +\lim _{n \rightarrow +\infty } \int _{\Omega } \varphi \left( x,\left| \nabla Z_{m}\left( u\right) \right| \right) \,dx}. \end{aligned}$$

(5.20)

Passing to the limit in (5.20) as $m \rightarrow +\infty ,$ we obtain

$$\begin{aligned} \lim _{m \rightarrow +\infty } \lim _{n \rightarrow +\infty } \int _{\Omega } F_{n}\left( x, u_{n}\right) \nabla Z_{m}\left( u_{n}\right) \,dx=0. \end{aligned}$$

Finally passing to the limit in (5.16), we get

$$\begin{aligned} \lim _{m \rightarrow +\infty } \lim _{n \rightarrow +\infty } \int _{\left\{ m \le u_{n} \le m+1 \right\} } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} \,dx=0. \end{aligned}$$

(5.21)

5.5 Almost everywhere convergence of the gradients

Let $v_{j} \in {\mathcal {D}}(\Omega )$ be a sequence such that $v_{j} \rightarrow u$ in $W_{0}^{1} L_{\varphi }(\Omega )$ for the modular convergence. For $m \ge k,$ we define the function $\varrho _{m}$ by

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

We denote by $\epsilon (n, \eta , j, m)$ all quantities (possibly different) such that

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

For fixed $k \ge 0,$ let $W_{\eta }^{n, j}=T_{\eta }\left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right)$ and $W_{\eta }^{j}=T_{\eta }\left( T_{k}(u)-T_{k}\left( v_{j}\right) \right) .$ Multiplying the approximating equation by $W_{\eta }^{n, j} \varrho _{m}\left( u_{n}\right)$, we obtain

$$\begin{aligned}&\displaystyle \int _{\Omega } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \nabla W_{\eta }^{n, j} \varrho _{m}\left( u_{n}\right) \,dx-\int _{\Omega } F_{n}\left( x, u_{n}\right) \nabla W_{\eta }^{n, j} \varrho _{m}\left( u_{n}\right) \,dx \nonumber \\&\quad \displaystyle \le \int _{\Omega } f_{n} W_{\eta }^{n, j} \varrho _{m}\left( u_{n}\right) d x+\int _{\Omega }\phi \nabla W_{\eta }^{n, j} \varrho _{m}\left( u_{n}\right) d x. \end{aligned}$$

(5.22)

Remark that if we take $n>m+1,$ we obtain

$$\begin{aligned} F_{n}\left( x, u_{n}\right) \varrho _{m}\left( u_{n}\right) =F\left( x, T_{m+1}\left( u_{n}\right) \right) \varrho _{m}\left( T_{m+1}\left( u_{n}\right) \right) , \end{aligned}$$

then $F_{n}\left( x, u_{n}\right)$ is bounded in $L_{\overline{\varphi }}(\Omega ),$ thus, by using the pointwise convergence of $u_{n}$ and Lebesgue’s theorem we obtain $F_{n}\left( x, u_{n}\right)$ converges to F(x, u) with the modular convergence as $n \rightarrow +\infty ,$ then

$$\begin{aligned} F_{n}\left( x, u_{n}\right) \varrho _{m}\left( u_{n}\right) \longrightarrow F(x, u) \varrho _{m}\left( u\right) \text { for } \sigma \left( \Pi L_{\varphi }, \Pi L_{\varphi }\right) . \end{aligned}$$

In the other hand for $0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta$ then $\nabla W_{\eta }^{n, j}=\nabla \Big (T_{k}(u_{n})-T_{k}(v_{j})\Big )$ converges to $\nabla \Big (T_{k}(u)- T_{k}(v_{j})\Big )$ weakly in $\left( L_{\varphi }(\Omega )\right) ^{N}$ as n tends to $+\infty$, then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow +\infty }\int _{\Omega } F_{n}\left( x, u_{n}\right) \varrho _{m}\left( u_{n}\right) \nabla W_{\eta }^{n, j} \,dx =\int _{\Omega } F(x, u) \varrho _{m}\left( u\right) \nabla W_{\eta }^{j} \,dx. \end{aligned}$$

By using the modular convergence of $W_{\eta }^{j}$ as $j \rightarrow +\infty$ and letting $\mu$ tends to infinity, we get

$$\begin{aligned} \int _{\Omega } F_{n}\left( x, u_{n}\right) \varrho _{m}\left( u_{n}\right) \nabla W_{\eta }^{n, j} \,dx=\epsilon (n, j) \quad \text{ for } \text{ any } m \ge 1. \end{aligned}$$

(5.23)

In the other hand for $n>m+1>k,$ we have $\nabla u_{n}\varrho ^{'}_{m}\left( u_{n}\right) =\nabla T_{m+1}\left( u_{n}\right)$ a.e. in $\Omega .$ By the almost every where convergence of $u_{n}$ we have $W_{\eta }^{n, j} \rightarrow W_{\eta }^{j}$ in $L^{\infty }(\Omega )$ weak- $^{*}$ and since the sequence $\left( F_{n}\left( x, T_{m+1}\left( u_{n}\right) \right) \right) _{n}$ converge strongly ${\text {in}} E_{\overline{\varphi }}(\Omega )$ then

$$\begin{aligned} F_{n}\left( x, T_{m+1}\left( u_{n}\right) \right) W_{\eta }^{n, j} \rightarrow F\left( x, T_{m+1}(u)\right) W_{\eta }^{j} \end{aligned}$$

converge strongly in $E_{\overline{\varphi }}(\Omega )$ as $n \rightarrow +\infty .$ By virtue of $\nabla T_{m+1}\left( u_{n}\right) \rightarrow \nabla T_{m+1}(u)$ weakly in $\left( L_{\varphi }(\Omega )\right) ^{N}$ as $n \rightarrow +\infty$ we have

$$\begin{aligned}&\displaystyle \lim _{n\rightarrow +\infty } \int _{\{m \le |u_{n}| \le m+1 \}} F_{n}(x, T_{m+1}(u_{n})) \nabla u_{n}\varrho ^{'}_{m}\left( u_{n}\right) W_{\eta }^{n, j} \,dx\displaystyle \nonumber \\&\quad =\int _{\{m \le |u| \le m+1\}} F(x, u) \nabla u \varrho ^{'}_{m}\left( u \right) W_{\eta }^{j} \,dx \end{aligned}$$

(5.24)

with the modular convergence of $W_{\eta }^{j}$ as $j \rightarrow +\infty$, we get

$$\begin{aligned} \int _{\Omega } F_{n}\left( x, u_{n}\right) \nabla u_{n} \varrho ^{'}_{m}\left( u_{n}\right) W_{\eta }^{n, j} \,dx=\epsilon (n, j) \quad \text{ for } \text{ any } m \ge 1 \end{aligned}$$

(5.25)

Concerning the first term of (5.22) we have

$$\begin{aligned}&\displaystyle \int _{\Omega } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \varrho ^{'}_{m}\left( u_{n}\right) W_{\eta }^{n, j} \,dx=\int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \varrho ^{'}_{m}\left( u_{n}\right) \nabla u_{n} W_{\eta }^{n, j} \,dx \nonumber \\&\quad \le \eta C \int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} \,dx, \end{aligned}$$

(5.26)

thus

$$\begin{aligned} \int _{\Omega } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \varrho ^{'}_{m}\left( u_{n}\right) W_{\eta }^{n, j} d x \le \epsilon (n, m). \end{aligned}$$

(5.27)

The weakly convergence of $T_{k}\left( u_{n}\right)$ to $T_{k}\left( v_{j}\right)$ in $W^{0,1} L_{\varphi }(\Omega )$ as n tends to $+\infty$, the bounded character of $W_{\eta }^{n, j}$, we obtain

$$\begin{aligned} \displaystyle \int _{\Omega } f_{n} \varrho _{m}\left( u_{n}\right) W_{\eta }^{n, j} \,dx = \epsilon (n, \eta ), \end{aligned}$$

(5.28)

and

$$\begin{aligned} \displaystyle \int _{\Omega } \phi \nabla W_{\eta }^{n, j}\varrho _{m}\left( u_{n}\right) \,dx = \epsilon (n,\eta ). \end{aligned}$$

(5.29)

Appealing now (1.5), we get

$$\begin{aligned}&\displaystyle \Big |\int _{\Omega } \phi \nabla u_{n} \varrho ^{'}_{m}\left( u_{n}\right) W_{\eta }^{n, j} \,dx\Big |\displaystyle \le \epsilon _{1} \nonumber \\&\quad \int _{\Omega } \overline{\varphi }\left( x, \frac{\phi }{\epsilon _{1}}\right) W_{\eta }^{n, j} \,dx+\epsilon _{1} \eta \int _{\{m \le \left| u_{n}\right| \le m+1\}} b_{n}\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} d x \le \epsilon (n, m, j, \eta ). \end{aligned}$$

(5.30)

In the other hand we have

$$\begin{aligned}&\displaystyle \int _{\Omega } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \varrho _{m}\left( u_{n}\right) \nabla W_{\eta }^{n, j} \,dx \nonumber \\&\quad \displaystyle =\int _{\left. \left\{ \left| u_{n}\right| \le k\right\} \cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \le \eta \right\} } b_{n}\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \varrho _{m}\left( u_{n}\right) \nonumber \\&\quad \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \right) \,dx \nonumber \\&\quad \displaystyle { -\int _{\left. \left\{ \left| u_{n}\right| >k\right\} \cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \le \eta \right\} } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \varrho _{m}\left( u_{n}\right) \nabla T_{k}\left( v_{j}\right) \,dx.} \end{aligned}$$

(5.31)

Since $b_{n}\left( x, T_{k+\eta }\left( u_{n}\right) , \nabla T_{k+\eta }\left( u_{n}\right) \right)$ is bounded in $\left( L_{\overline{\varphi }}(\Omega )\right) ^{N},$ there exist some $\varpi _{k+\eta } \in \left( L_{\overline{\varphi }}(\Omega )\right) ^{N}$ such that $b_{n}\left( x, T_{k+\eta }\left( u_{n}\right) , \nabla T_{k+\eta }\left( u_{n}\right) \right) \rightharpoonup \varpi _{k+\eta }$ weakly in $\left( L_{\overline{\varphi }}(\Omega )\right) ^{N}.$ Thus:

$$\begin{aligned} \begin{array}{l}\displaystyle { \int _{\left. \left\{ \left| u_{n}\right|>k\right\} \cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \le \eta \right\} } b_{n}\left( x, u_{n}, \nabla u_{n}\right) \varrho _{m}\left( u_{n}\right) \nabla T_{k}\left( v_{j}\right) \,dx } \\ \displaystyle { =\int _{\left. \{|u|>k\} \cap \left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \right) \le \eta \right\} } \varrho _{m}(u) \varpi _{k+\eta }\nabla T_{k}\left( v_{j}\right) \,dx+\epsilon (n)}, \end{array} \end{aligned}$$

(5.32)

By letting $j \rightarrow +\infty ,$ we get

$$\begin{aligned} \displaystyle \int _{\left. \{|u|>k\} \cap \left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \right) \le \eta \right\} } \varrho _{m}(u) \nabla T_{k}\left( v_{j}\right) \varpi _{k+\eta } \,dx \displaystyle =\int _{\left. \{|u|>k\}\right\} } \varrho _{m}(u) \nabla T_{k}(u) \varpi _{k+\eta } d x+\epsilon (n, j) =\epsilon (n, j). \end{aligned}$$

(5.33)

Thanks to (5.23)–(5.33), one has

$$\begin{aligned}&\int _{\left. \left\{ \left| u_{n}\right| \le k\right\} \cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \le \eta \right\} } b_{n}\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \varrho _{m}\left( u_{n}\right) \nonumber \\&\quad \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \right) \,dx \le C \eta +\epsilon (n, j, m). \end{aligned}$$

(5.34)

Since $\exp \left( G\left( u_{n}\right) \right) \ge 1$ and $\varrho _{m}\left( u_{n}\right) =1$ for $\left| u_{n}\right| \le k$ then

$$\begin{aligned}&\displaystyle \int _{ \{|u_{n}| \le k\} \cap \{0 \le T_{k}(u_{n})-T_{k}(v_{j}) \le \eta \}} b_{n}\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \right) \,dx \nonumber \\&\quad \le C \eta +\epsilon (n, j, m). \end{aligned}$$

(5.35)

Finally we show that,

$$\begin{aligned} \displaystyle { \int _{\Omega }\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \Big )\Big (\nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\Big ) \,dx \rightarrow 0.} \end{aligned}$$

(5.36)

For $s>0,$ denoting by $\Omega ^{s}=\left\{ x \in \Omega :\left| \nabla T_{k}(u)\right| \le s\right\}$ and $\Omega _{j}^{s}=\left\{ x \in \Omega :\left| \nabla T_{k}\left( v_{j}\right) \right| \le \right.$ $s\}$ then by $\chi ^{s}$ and $\chi _{j}^{s}$ the characteristic functions of $\Omega ^{s}$ and $\Omega _{j}^{s}$ respectively, letting $0<\delta <1$, define

$$\begin{aligned} \Theta _{n, k}=\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \Big )\Big (\nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\Big ). \end{aligned}$$

For $s>0,$ we have

$$\begin{aligned} 0 \le \int _{\Omega ^{s}} \Theta _{n, k}^{\delta } \,dx=\int _{\Omega ^{s}} \Theta _{n, k}^{\delta } \chi _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } d x+\int _{\Omega ^{s}} \Theta _{n, k}^{\delta } \chi _{\left\{ T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) >\eta \right\} } \,dx. \end{aligned}$$

The first term of the right-side hand, with the Hölder inequality we obtain

$$\begin{aligned}&\displaystyle \int _{\Omega ^{s}} \Theta _{n, k}^{\delta } \chi _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } \,dx \le \left( \int _{\Omega ^{*}} \Theta _{n, k} \chi _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } d x\right) ^{\delta }\left( \int _{\Omega ^{*}} \,dx\right) ^{1-\delta } \nonumber \\&\quad \displaystyle \le C_{1}\left( \int _{\Omega ^{s}} \Theta _{n, k} \chi _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } \,dx\right) ^{\delta }. \end{aligned}$$

(5.37)

For the second term of the right-side hand by the Hölder inequality we have

$$\begin{aligned} \int _{\Omega ^{s}} \Theta _{n, k}^{\delta } \chi _{\left\{ T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right)>\eta \right\} } d x \le \left( \int _{\Omega ^{s}} \Theta _{n, k} \,dx\right) ^{\delta }\left( \int _{\{T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) >\eta \}} \,dx\right) ^{1-\delta }, \end{aligned}$$

(5.38)

since $a\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right)$ is bounded in $\left( L_{\overline{\varphi }}(\Omega )\right) ^{N},$ while $\nabla T_{k}\left( u_{n}\right)$ is bounded in $\left( L_{\varphi }(\Omega )\right) ^{N}$ then

$$\begin{aligned} \int _{\Omega ^{s}} \Theta _{n, k}^{\delta } \chi _{\left\{ T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right)>\eta \right\} } \,dx \le C_{2} {\text {meas}}\left\{ x \in \Omega : T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) >\eta \right\} ^{1-\delta } \end{aligned}$$

(5.39)

We obtain

$$\begin{aligned}&\displaystyle \int _{\Omega ^{g}} \Theta _{n, k}^{\delta } d x \le C_{1}\left( \int _{\Omega ^{s}} \Theta _{n, k} \chi _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } \,dx\right) ^{\delta }\nonumber \\&\quad +C_{2} {\text {meas}}\left\{ x \in \Omega : T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) >\eta \right\} ^{1-\delta } \end{aligned}$$

(5.40)

On the other hand

$$\begin{aligned}&\displaystyle \displaystyle \int _{\Omega ^{s}} \Theta _{n, k} \chi _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } \,dx \nonumber \\&\quad \displaystyle \le \int _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} }\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi _{s}\right) \Big ) \nonumber \\&\qquad \times \Big (\nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi _{s}\Big )\,dx. \end{aligned}$$

(5.41)

For each $s,\ r\in {\mathbb {R}}^+$ with $s>r$ one has

$$\begin{aligned}&\displaystyle 0 \le \int _{\left. \Omega ^{r}\cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right) \right\} }\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \Big ) \nonumber \\&\qquad \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) \,dx \nonumber \\&\quad \displaystyle \le \int _{\left. \Omega ^{s} \cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right) \right\} }\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \Big ) \nonumber \\&\qquad \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) \,dx \nonumber \\&\quad \displaystyle =\int _{\Omega ^{s}\cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} }\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi _{s}\right) \Big ) \nonumber \\&\qquad \displaystyle \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi _{s}\right) \,dx \nonumber \\&\quad \displaystyle \le \int _{\left. \Omega \cap \left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right) \right\} }\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \Big ) \nonumber \\&\qquad \displaystyle \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u) \chi ^{s}\right) \,dx \nonumber \\&\quad \displaystyle =\int _{\{0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \}}\Big (b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \Big ) \nonumber \\&\qquad \displaystyle \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \,dx \nonumber \\&\qquad \displaystyle +\int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}(u) \chi ^{s}\right) \,dx \nonumber \\&\qquad \displaystyle +\int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} }\left( b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \right) \nabla T_{k}\left( u_{n}\right) \,dx \nonumber \\&\qquad \displaystyle -\int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}) \,dx \nonumber \\&\qquad \displaystyle +\int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u) \chi ^{s}\right) \nabla T_{k}(u) \chi ^{s}) \,dx \nonumber \\&\quad \displaystyle =I_{1}+I_{2}+I_{3}+I_{4}+I_{5}. \end{aligned}$$

(5.42)

In the sequel we pass to the limit in $I_i$ when $n,\ j,\ \mu ,$ and $s \rightarrow +\infty$. We have

$$\begin{aligned}&I_{1}=\int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \right) d x \\&\begin{array}{llll}&{}\displaystyle { -\int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\Big (x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \Big )\left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}\left( v_{j}\right) \right) d x } \\ &{}\quad \displaystyle { -\int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) d x } \end{array} \end{aligned}$$

Thanks to (5.35), the first term of the right hand side in $I_1$, we get

$$\begin{aligned}&\displaystyle { \int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \right) d x} \\&\begin{array}{clll}&{}\displaystyle { \le C \eta +\epsilon (n, m, j, s)-\int _{\left\{ |u|>k \cap 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}(u), 0\right) \nabla T_{k}\left( v_{j}\right) \,dx } \\ &{}\quad \le C \eta +\epsilon (n, m, j). \end{array} \end{aligned}$$

Since $b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right)$ is bounded in $\left( L_{\overline{\varphi }}(\Omega )\right) ^{N},$ there exist some $\varpi _{k} \in$ $\left( L_{\overline{\varphi }}(\Omega )\right) ^{N}$ such that (for a subsequence still denoted by $u_{n}$):

$$\begin{aligned}&b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \\&\quad \rightarrow \varpi _{k} \quad \text{ in } \quad \left( L_{\varphi }(\Omega )\right) ^{N} \quad \text{ for } \quad \sigma \left( \Pi L_{\varphi }, \Pi E_{\varphi }\right) \end{aligned}$$

By using in the fact $\left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}\left( v_{j}\right) \right) \chi _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} }$ strongly converges to $\left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}\left( v_{j}\right) \right) \chi _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} }$ in $\left( E_{\varphi }(\Omega )\right) ^{N}$ as $n \rightarrow +\infty$.

The second term of the right hand side of $I_1$ tends to

$$\begin{aligned}&\displaystyle \int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\Big (x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \Big )\left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}\left( v_{j}\right) \right) \,dx \\&\quad =\int _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } \varpi _{k}\left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}\left( v_{j}\right) \right) \,dx + \epsilon (n). \end{aligned}$$

The third term of the right-hand side tends to

$$\begin{aligned}&\displaystyle \int _{\left\{ 0 \le T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \,dx \\&\quad =\int _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}(u), \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \left( \nabla T_{k}(u)-\nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}\right) \,dx+\epsilon (n), \end{aligned}$$

Letting $j \rightarrow +\infty$ and $\mu \rightarrow +\infty$ of $I_1$, it possible to conclude that

$$\begin{aligned} I_{1} \le C \eta +\epsilon (n, j, s). \end{aligned}$$

Concerning $I_{2},$ by letting $n \rightarrow +\infty ,$ we obtain

$$\begin{aligned} I_{2} \rightarrow \int _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } \varpi _{k}\left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}(u) \chi ^{s}\right) \,dx. \end{aligned}$$

Since $b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \rightarrow \varpi _{k}$ in $\left( L_{\overline{\varphi }}(\Omega )\right) ^{N},$ for $\sigma \left( \Pi L_{\overline{\varphi }}, \Pi E_{\varphi }\right)$ while

$$\begin{aligned} \left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}(u) \chi ^{s}\right) \chi _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } \rightarrow \left( \nabla T_{k}\left( v_{j}\right) \chi _{j}^{s}-\nabla T_{k}(u) \chi ^{s}\right) \chi _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } \end{aligned}$$

strongly in $\left( E_{\varphi }(\Omega )\right) ^{N}$. Now, letting $j \rightarrow +\infty ,$ and thanks to Lebesgue’s theorem, we obtain

$$\begin{aligned}&I_{2}=\epsilon (n, j), \\&I_{3}=\epsilon (n, j), \\&\displaystyle {I_{4}=\int _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}(u), \nabla T_{k}(u)\right) \nabla T_{k}(u) d x+\epsilon (n, j, s, m)}, \end{aligned}$$

and

$$\begin{aligned} \displaystyle I_{5}=\int _{\left\{ 0 \le T_{k}(u)-T_{k}\left( v_{j}\right) \le \eta \right\} } b\left( x, T_{k}(u), \nabla T_{k}(u)\right) \nabla T_{k}(u) d x+\epsilon (n, j, s, m). \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} \int _{\Omega ^{s}} \Theta _{n, k} d x \le C_{1}(C \eta +\epsilon (n, \eta , m))^{\delta }+C_{2}(\epsilon (n,))^{1-\delta }. \end{aligned}$$

Which leads to

$$\begin{aligned}&\displaystyle \int _{\left\{ T_{\eta }\left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \ge 0\right\} \cap \Omega ^{r}} \Big [\Big (b(x,T_{k}(u_{n}), \nabla T_{k}(u_{n}))-b(x,T_{k}(u_{n}), \nabla T_{k}(u))\Big ) \nonumber \\&\quad \times (\nabla T_{k}(u_{n})-\nabla T_{k}(u))\Big ]^{\delta } \,dx=\epsilon (n). \end{aligned}$$

(5.43)

By taking $W_{\eta }^{n,j}=T_{\eta }\left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) ^{-}$ and $W_{\eta }^{j}=T_{\eta }\left( T_{k}(u)-T_{k}\left( v_{j}\right) \right) ^{-},$ then testing the approximating equation by $\exp \left( G\left( u_{n}\right) \right) W_{\eta }^{n, j} \varrho _{m}\left( u_{n}\right) ,$ we obtain

$$\begin{aligned}&\int _{\left\{ T_{\eta }\left( T_{k}\left( u_{n}\right) -T_{k}\left( v_{j}\right) \right) \le 0\right\} \cap \Omega ^{r}}\Big [\Big (b\left( x,T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x,T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \Big ) \nonumber \\&\quad \times \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) \Big ]^{\delta } \,dx=\epsilon (n). \end{aligned}$$

(5.44)

Thanks to (5.43) and (5.44) we have

$$\begin{aligned} \int _{\Omega ^{r}}\Big [\left( b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) -b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}(u)\right) \right) \left( \nabla T_{k}\left( u_{n}\right) -\nabla T_{k}(u)\right) \Big ]^{\delta } \,dx=\epsilon (n) \end{aligned}$$

As a consequence, since r is arbitrary:

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

(5.45)

and for all $k \ge 0$, we have

$$\begin{aligned}&b\left( x, T_{k}\left( u_{n}\right) , \nabla T_{k}\left( u_{n}\right) \right) \rightharpoonup b\left( x, T_{k}(u), \nabla T_{k}(u)\right) \text { weakly in }\left( L_{\psi }(\Omega )\right) ^{N}, \end{aligned}$$

(5.46)

$$\begin{aligned}&\varphi \left( x,\left| \nabla T_{k}\left( u_{n}\right) \right| \right) \rightarrow \varphi \left( x,\left| \nabla T_{k}(u)\right| \right) \text { strongly in } L^{1}(\Omega ). \end{aligned}$$

(5.47)

5.6 Renormalization identity for the solutions

We show that The limit u of the approximate solution $u_{n}$ of (5.1) satisfies:

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

(5.48)

To this end, remark that for any $m>0$ one has

$$\begin{aligned}&\displaystyle \int _{\left\{ m \le \left| u_{n}\right| \le m+1\right\} } b\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} \,dx \displaystyle =\int _{\Omega } b\left( x, u_{n}, \nabla u_{n}\right) \left( \nabla T_{m+1}\left( u_{n}\right) -\nabla T_{m}\left( u_{n}\right) \right) \,dx \nonumber \\&\quad \displaystyle =\int _{\Omega } b\Big (x, T_{m+1}\left( u_{n}\right) , \nabla T_{m+1}\left( u_{n}\right) \Big ) \nabla T_{m+1}\left( u_{n}\right) \,dx \displaystyle \nonumber \\&\quad -\int _{\Omega } b\Big (x, T_{m}\left( u_{n}\right) , \nabla T_{m}\left( u_{n}\right) \Big ) \nabla T_{m}\left( u_{n}\right) \,dx. \end{aligned}$$

(5.49)

According to (5.46), (5.47) one is at liberty to pass to the limit as n tends to infinity for fixed m and to obtain

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

(5.50)

Taking the limit as m tends to $+\infty$ and using the estimate (5.21) show that u satisfies (5.48).

5.7 Passing to the limit

Let $h \in {\mathcal {C}}_{c}^{1}({\mathbb {R}})$ and $V \in {\mathcal {D}}(\Omega ).$ Using the admissible test function $h\left( u_{n}\right) V$ in (5.1) leads to

$$\begin{aligned} \begin{array}{l}\displaystyle { \int _{\Omega } b\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} h^{\prime }\left( u_{n}\right) V d x+\int _{\Omega } b\left( x, u_{n}, \nabla u_{n}\right) \nabla V h\left( u_{n}\right) d x }\\ \displaystyle { \quad +\int _{\Omega } F_{n}\left( x, u_{n}\right) \nabla \left( h\left( u_{n}\right) V\right) d x }\displaystyle { =\int _{\Omega } f_{n} h\left( u_{n}\right) V d x + \int _{\Omega } \phi \nabla \left( h\left( u_{n}\right) V\right) d x . } \end{array}\end{aligned}$$

(5.51)

We shall pass to the limit in each term in the previous equality, to this end, remark that since h and $h^{\prime }$ have a compact support in h,  there exists $K>0$ such that $supp(h)\subset [-K, K].$ For n large enough, we have:

$$\begin{aligned} \begin{aligned} F_{n}(x, t) h(t)&=F_{n}\left( x, T_{n}(t)\right) h(t)=F\left( x, T_{K}(t)\right) h(t) \\ F_{n}(x, t) h^{\prime }(t)&=F_{n}\left( x, T_{n}(t)\right) h^{\prime }(t)=F\left( x, T_{K}(t)\right) h^{\prime }(t) \end{aligned} \end{aligned}$$

Let us start by the third integral of the left-hand side and the right hand-side of (5.51). Since $h \in C _{c}^{1}({\mathbb {R}})$ and $V \in {\mathcal {D}}(\Omega ),$ then there exists two positive constants $c_{1}$ and $c_{1}^{\prime }$ such that $\left\| h\left( T_{K}\left( u_{n}\right) \right) \nabla V\right\| _{\infty } \le c_{1}$ and $\left\| h^{\prime }(t)\left( T_{K}\left( u_{n}\right) V \nabla T_{K}\left( u_{n}\right) \Vert _{\infty } \le c_{1}^{\prime }\right. \right.$ Now since $T_{K}\left( u_{n}\right)$ is bounded in $W_{0}^{1} L_{\varphi }(\Omega ),$ then there exists two positive constant $\lambda _{0}$ and $\lambda$ such that $\displaystyle {\int _{\Omega } \varphi \left( x,\frac{\left| \nabla T_{K}\left( u_{n}\right) \right| }{\lambda }\right) d x }\le \lambda _{0} .$ Using the convexity and monotonicity of $\varphi ,$ for $\eta$ large enough, we can write

$$\begin{aligned} \begin{array}{lllll} &{}\displaystyle { \int _{\Omega } \varphi \left( x,\frac{\nabla \left( h\left( T_{K}\left( u_{n}\right) \right) V\right) }{\eta }\right) d x }\\ &{}\quad \displaystyle { =\int _{\Omega } \varphi \left( x,\frac{h\left( T_{K}\left( u_{n}\right) \right) \nabla V+h^{\prime }(t)\left( T_{K}\left( u_{n}\right) V\left| \nabla T_{K}\left( u_{n}\right) \right| \right. }{\eta }\right) d x} \\ &{}\quad \displaystyle \le \int _{\Omega } \varphi \left( x,{ c_{1}+{c_{1}^{\prime } \lambda {\left| \nabla T_{K}(u_{n})\right| \over \lambda }} \over \eta }\right) \,dx \\ &{}\quad \displaystyle { \le \int _{\Omega } \varphi \left( x,\frac{c_{1}}{\eta }\right) \,dx+\frac{c_{1}^{\prime } \lambda }{\eta } \int _{\Omega } \varphi \left( x,\frac{\left| \nabla T_{K}\left( u_{n}\right) \right| }{\lambda }\right) \,dx} \\ &{}\quad \displaystyle { \le C_{\eta , c_{1}}+\frac{c_{1}^{\prime } \lambda \lambda _{0}}{\eta } \quad \text{ where } C_{\eta , c_{1}}=\int _{\Omega } \varphi \left( x,\frac{c_{1}}{\eta }\right) d x<\infty }. \end{array} \end{aligned}$$

Then the sequence $\left\{ \nabla \left( h\left( T_{K}\left( u_{n}\right) \right) V\right) \right\}$ is bounded in $\left( L_{\varphi }(\Omega )\right) ^{N},$ as a consequence, we deduce

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

(5.52)

Moreover, since $F\left( x, T_{K}\left( u_{n}\right) \right)$ is bounded in $L_{\psi }(\Omega ),$ we have from Lemma 3.10

$$\begin{aligned} F\left( x, T_{K}\left( u_{n}\right) \right) \rightarrow F\left( x, T_{K}(u)\right) \quad \text{ strongly } \text{ in } E_{\psi }(\Omega ). \end{aligned}$$

By (5.52), we get

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega } F_{n}\left( x, u_{n}\right) \nabla \left( h\left( u_{n}\right) V\right) d x=\int _{\Omega } F\left( x, T_{K}(u)\right) \nabla (h(u) V) d x. \end{aligned}$$

Moreover we have

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega } f_{n} h\left( u_{n}\right) V d x=\int _{\Omega } f h(u) V d x, \\\lim _{n \rightarrow \infty } \int _{\Omega } \phi \nabla h\left( u_{n}\right) V d x = \int _{\Omega } \phi \nabla h(u)V d x .\end{aligned}$$

Concerning the first integral of (5.51), while supp $h^{\prime } \subset [-K, K],$ we obtain

$$\begin{aligned} h^{\prime }\left( u_{n}\right) V b\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} =h^{\prime }\left( u_{n}\right) V b\left( x, T_{K}\left( u_{n}\right) , \nabla T_{K}\left( u_{n}\right) \right) \nabla T_{K}\left( u_{n}\right) \quad \text{ a.e. } \text{ in } \Omega . \end{aligned}$$

The pointwise convergence of $u_{n}$ to u, the bounded character of $h'V$, (5.46) and (5.47) imply that

$$\begin{aligned} h^{\prime }\left( u_{n}\right) V b\left( x, u_{n}, \nabla u_{n}\right) \nabla u_{n} \rightharpoonup h^{\prime }(u) V b\left( x, T_{K}(u), \nabla T_{K}(u)\right) \nabla T_{K}(u) \text { weakly in } L^{1}(\Omega ). \end{aligned}$$

The term $h^{\prime }(u) V b\left( x, T_{K}(u), \nabla T_{K}(u)\right) \nabla T_{K}(u)$ is identified with $h^{\prime }(u) V b\left( x, u, \nabla u\right) \nabla u$.

Now since $h\left( u_{n}\right) V b\left( x, u_{n}, \nabla u_{n}\right) =h\left( u_{n}\right) V b\left( x, T_{K}\left( u_{n}\right) , \nabla T_{K}\left( u_{n}\right) \right)$ a.e. in $\Omega$, and using the strongly convergence of $h\left( u_{n}\right) \nabla V$ to $h(u) \nabla V$ in $\left( E_{\varphi }(\Omega )\right) ^{N},$ and using the weakly convergence of $b\left( x, T_{K}\left( u_{n}\right) , \nabla T_{K}\left( u_{n}\right) \right)$ to $b\left( x, T_{K}(u), \nabla T_{K}(u)\right)$ in $\left( L_{\psi }(\Omega )\right) ^{N}$ for $\sigma \left( \Pi L_{\psi }, \Pi E_{\varphi }\right) ,$ then

$$\begin{aligned} \displaystyle \lim _{n \rightarrow \infty } \int _{\Omega } b\left( x, u_{n}, \nabla u_{n}\right) \nabla V h\left( u_{n}\right) d x=\int _{\Omega } b(x, u, \nabla u) \nabla V h(u) \,dx. \end{aligned}$$

As a consequence of the above convergence results, we are in a position to pass to the limit as n tends to $+\infty$ in (5.51) and to conclude that u satisfies (4.3). As a conclusion of Step 5.1 to Step 5.7, the proof of Theorem 4.1 is complete.

Remark 6

• (1) It is possible to extend this result to the following parabolic equation

  $$\begin{aligned} {\left\{ \begin{array}{ll}\frac{\partial u}{\partial t}-{\text {div}}(a(x, t, u, \nabla u))+ F(x, t, u)=\mu &{} \text{ in } \Omega \times (0, T), \\ u=0 &{} \text{ on } \partial \Omega \times (0, T), \\ u(x, 0)=u_0(x) &{} \text{ in } \Omega .\end{array}\right. } \end{aligned}$$

  where $\Omega$ is a bounded open subset of ${\mathbb {R}}^N, N \ge 1, T>0$ and $Q_T$ is the cylinder $\Omega \times (0, T)$. The operator $A(u)=-{\text {div}}(a(x, t, u, \nabla u))$ is a Leray-Lions operator lefined in $W_0^{1, x} L_\varphi \left( Q_T\right)$. The lower order term F verifies the natural growth condition, no $\Delta _{2}$-condition is assumed on the Musielak function, and the datum $\mu$ is assumed to belong to $L^{1}(Q_T)+W^{-1} E_{\psi }(Q_T)$.

• (2) In the case of $F\equiv 0,$ the problem (1.1) admits a unique solution.