Open Access

Existence of solutions for a class of porous medium type equations with lower order terms

Journal of Inequalities and Applications20152015:294

https://doi.org/10.1186/s13660-015-0799-9

Received: 5 May 2015

Accepted: 27 August 2015

Published: 18 September 2015

Abstract

This paper deals with a class of degenerate quasilinear elliptic equations of the form \(-\operatorname{div}(a(x,u,\nabla u))+F(x,u,\nabla u)=f \), where \(a(x,u,\nabla u)\) is allowed to degenerate with the unknown u. Under some hypothesis on a, F, and f, we obtain the existence of bounded solutions \(u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). For the case \(f\in L^{1}(\Omega)\), we also prove that there exists at least one renormalized solution.

Keywords

degenerate equations weak and renormalized solutions \(L^{\infty}\) estimate natural growth

MSC

35D05 35J60 35J70 26D07

1 Introduction

This paper concerns the following degenerate problem:
$$(\mathscr{P})\quad \left \{ \textstyle\begin{array}{l@{\quad}l} -\operatorname{div}(a(x,u,\nabla u))+F(x,u,\nabla u)=f &\mbox{in } \Omega, \\ u=0 &\mbox{on } \partial\Omega, \end{array}\displaystyle \right . $$
where Ω is a bounded domain of \(\mathbb{R}^{N}\) (\(N\geq2\)), \(f\in L^{q}(\Omega)\) with \(q\geq1\) and \(a(x,s,\xi)\) is a Carathéodory function. Furthermore, we assume that there exists a continuous function α from \(\mathbb{R}^{+}\) into \(\mathbb{R}^{+}\) such that \(\alpha(0)=0\) and \(a(x,s,\xi)\xi\geq\alpha(|s|)|\xi|^{p}\) for any \(s\in\mathbb{R}\), \(\xi \in \mathbb{R}^{N}\), and almost every x in Ω. Thus problem (\(\mathscr{P}\)) degenerates for the subset \(\{x\in\Omega:u(x)=0\}\).
Problem (\(\mathscr{P}\)) has important and extensive applications to the fluid dynamics in porous media, in hydrology and in petroleum engineering (see [1, 2]). The simplest model is the stationary case of the porous media equation with zero Dirichlet boundary condition:
$$(\mathrm{P}_{0})\quad \left \{ \textstyle\begin{array}{l@{\quad}l} -\triangle(|u|^{m-1}u)+F(x,u)=f &\mbox{in } \Omega, \\ u=0 &\mbox{on } \partial\Omega, \end{array}\displaystyle \right . $$
which has been widely studied in the literature (see [36] and references therein).

For the case \(\alpha\equiv\mathrm{constant}>0\), the existence of bounded solutions to problem (\(\mathscr{P}\)) is proved in [7], when the data f is small in a suitable norm.

Concerning the case that α is a positive function, Porretta and Segura de León investigated the existence results to problem (\(\mathscr{P}\)); see [8]. We remark that in [8], no sign condition is imposed on F, but the growth of F at infinity need to be controlled. We also point out that a variational inequality related to problem (\(\mathscr{P}\)) was studied in [9], and similar results can be found in [10] and [11].

In the case \(\alpha(0)=0\), \(f\in W^{-1,r}(\Omega)\cap L^{1}(\Omega)\) with \(r\geq p'\), \(r>\frac{N}{p-1} \), Rakotoson proved the existence of a bounded weak solution to problem (\(\mathscr{P}\)) (see [12]), provided that F satisfies a sign condition. As \(F=0\) and \(f\in W^{-1,r}(\Omega)\), the existence of solutions to problem (\(\mathscr{P}\)) has been discussed in [13]. We point out that the parabolic version of [13] has been studied in [14].

As \(f\in L^{q}(\Omega)\) with \(q\geq\max\{1,\frac{N}{p}\}\), we shall give a direct method to prove the existence of bounded weak solutions to problem (\(\mathscr{P}\)) in the standard sense, i.e. \(u\in W_{0}^{1,p}(\Omega)\). The main difficulty comes from the facts that its modulus of ellipticity vanishes when the solution u vanishes. To overcome this difficulty, we shall firstly establish the \(L^{\infty}\) estimate for solution u, by the technique of rearrangement which is differs from the usual Stampacchia \(L^{\infty}\) regularity procedure. Then, by constructing suitable approximate problems, and using a priori estimates and a test function method, we shall finish the proof of this existence results.

Furthermore, we will study the case when \(f\in L^{1}(\Omega)\). Since no growth conditions are required for ω and β (see (H2)), it is not obvious that the term \(-\operatorname{div}(a(x, u, \nabla u))\) makes sense even as a distribution. To overcome this difficulty, we shall use the concept of renormalized solutions, which is introduced by Diperna and Lions (see [15]). This notion was adapted by many authors to study partial differential equations with measurable data, especially for \(L^{1}\) data (see [1618] for example). We remark that an equivalent notion called entropy solutions, was introduced independently by Bénilan et al. [19].

The main ideas and methods come from [8, 10, 12, 20]. This paper is organized as follows: in Section 2 we give some preliminaries and state the main results; in Section 3, we study the existence of bounded solution to problem (\(\mathscr{P}\)); in Section 4, we prove the existence of renormalized solution.

2 Some preliminaries and the main results

2.1 Properties of the relative rearrangement

Let Ω be a bounded open subsets of \(\mathbb{R}^{N}\), we denote by \(|E|\) the Lebesgue measure of a set E. Assume that \(u: \Omega \rightarrow\mathbb{R}\) be a measurable function, we define the distribution function \(\mu_{u}(t)\) of u as follows:
$$\mu_{u}(t)=\bigl\vert \bigl\{ x\in\Omega: u(x)>t\bigr\} \bigr\vert ,\quad \forall t\in\mathbb{R}. $$
The decreasing rearrangement \(u_{\ast}\) of u is defined as the generalized inverse function of \(\mu_{u}(t)\), i.e.
$$u_{\ast}(s)=\inf\bigl\{ t\in R: \mu_{u}(t)\leq s\bigr\} , \quad s\in\Omega^{\ast }=\bigl[0,\vert \Omega \vert \bigr]. $$
We recall also that u and \(u_{\ast}\) are equi-measurable, i.e.
$$\mu_{u}(t)=\mu_{u_{\ast}}(t),\quad t\in\mathbb{R}, $$
which implies that for any non-negative Borel function ψ we have
$$ \int_{\Omega}\psi\bigl(u(x)\bigr)\, \mathrm{d}x=\int _{0}^{|\Omega|}\psi\bigl(u_{\ast }(s)\bigr)\, \mathrm{d}s, $$
and if \(E\subset\Omega\) be a measurable subset, then
$$ \int_{E}u(x)\, dx\leq\int_{0}^{|E|}u_{\ast}(s) \, ds. $$

Using the Fleming-Rishel formula, Hölder’s inequality, and the isoperimetric inequality, we can get the following result (see [7, 9, 12]).

Lemma 2.1

For any non-negative function \(u\in W_{0}^{1,1}(\Omega)\), the following chain of inequalities holds:
$$ NC_{N}^{1/N}\mu_{u}(t)^{1-1/N}\leq- \frac{d}{dt}\int_{u>t}|\nabla u|\, \mathrm{d}x\leq \bigl(-\mu_{u}'(t)\bigr)^{1/p'}\biggl(- \frac{d}{dt}\int_{u>t}|\nabla u|^{p}\, \mathrm{d}x\biggr)^{1/p}, $$
where \(C_{N}\) denotes the measure of the unit ball in \(\mathbb{R}^{N}\).

For more details as regards the theory of rearrangement, we just refer to [21] and the references therein.

2.2 Assumptions and the main results

Let Ω be an open bounded set of \(\mathbb{R}^{N}\) (\(N\geq2\)) and \(p>1\), we make the following assumptions.
(H1): 
\(a: \Omega\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}\) is a Carathéodory vector function satisfying: there exists a continuous function α from \(\mathbb{R}_{+}\) into \(\mathbb {R}_{+}\) such that \(\alpha(0)=0\) and \(\alpha(s)>0\) if \(s>0\) and
$$\begin{aligned}& a(x,s,\xi)\xi\geq\alpha\bigl(\vert s\vert \bigr)|\xi|^{p},\quad \forall s\in R, \mbox{a.e. } x\in \Omega, \forall\xi\in\mathbb{R}^{N}, \\& \int_{0}^{+\infty}\alpha^{\frac{1}{p-1}}(s)\, \mathrm{d}s=\int_{0}^{+\infty }\frac{1}{\alpha(s)}\, \mathrm{d}s =+\infty \end{aligned}$$
and
$$ \frac{1}{\alpha}\in L^{1}(0,b) \quad \mbox{for any given }b>0. $$
(H2): 
There exists a Carathéodory vector function ā such that for a.e. \(x\in\Omega\), \(\forall s\in\mathbb{R}\), \(\forall\xi, \xi'\in \mathbb{R}^{N}\) with \(\xi\neq\xi'\):
  1. (i)

    \(a(x,s,\xi)=\alpha(|s|)\bar{a}(x,s,\xi)\).

     
  2. (ii)

    \([\bar{a}(x,s,\xi)-\bar{a}(x,s,\xi')][\xi-\xi']>0\).

     
  3. (iii)
    There exist an increasing function ω from \(\mathbb{R}^{+}\) into \(\mathbb{R}^{+}\) and a non-negative function \(\bar{\omega}\in L^{p'}(\Omega)\) such that
    $$\bigl\vert \bar{a}(x,s,\xi)\bigr\vert \leq\omega\bigl(\vert s\vert \bigr)\bigl[|\xi|^{p-1}+\bar{\omega}(x)\bigr]. $$
     
  4. (iv)
    The function ā is a positively homogeneous of degree \((p-1)\) with respect to the variable ξ, i.e.
    $$\bar{a}(x,s,t\xi)=t^{p-1}\bar{a}(x,s,\xi),\quad \forall t\geq 0. $$
     
(H3): 
\(F:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow \mathbb{R}\) is a Carathéodory function, for which there exists an increasing function β from \([0,+\infty)\) into \([0,+\infty)\) vanishing and continuous at zero such that for a.e. \(x\in\Omega\), \(\forall s\in\mathbb{R}\) and \(\forall\xi\in\mathbb {R}^{N}\):
$$\bigl\vert F(x,s,\xi)\bigr\vert \leq\beta\bigl(\vert s\vert \bigr)| \xi|^{p}. $$
(H4): 

\(f\in L^{q}(\Omega)\) with \(q> \max\{1,\frac{N}{p}\}\).

(H5): 
\(\lim_{s\rightarrow \infty}\frac{e^{\gamma(|s|)}}{(1+\phi(|s|))^{p-1}}=0\), where γ and ϕ are defined as follows:
$$ \gamma(s)=\int_{0}^{s} \frac{\beta(|\sigma|)}{\alpha(|\sigma|)}\, \mathrm {d}\sigma;\qquad \phi(s)=\int_{0}^{s} \bigl(\alpha\bigl(|\sigma|\bigr)\bigr)^{\frac{1}{p-1}}e^{\frac{\gamma (|s|)}{p-1}}\, \mathrm{d}\sigma. $$
(2.1)

Remark 2.1

Assumption (H1) allows us to consider the porous medium operators \(\triangle(|u|^{m-1}u)=\operatorname {div}(m|u|^{m-1}\nabla u)\). In this case, it yields \(\alpha (|s|)=|s|^{m-1}\), so that the conditions \(\alpha(0)=0\) and \(\frac {1}{\alpha}\in L^{1}(0,b)\) indicate \(1< m<2\). Thus, in this case, the porous medium equation becomes a slow diffusion equation.

We now introduce several auxiliary functions by
$$\begin{aligned}& \tilde{\alpha}(s)=\int_{0}^{s} \alpha^{\frac{1}{p-1}}\bigl(\vert t\vert \bigr)\,\mathrm{d}t, \end{aligned}$$
(2.2)
$$\begin{aligned}& \gamma_{\theta}(s)=\int_{0}^{s} \frac{\beta (|\sigma|)}{\alpha(|\sigma|)+\theta}\,\mathrm{d}\sigma \quad \mbox{for any fixed } \theta>0, \end{aligned}$$
(2.3)
$$\begin{aligned}& \tilde{\gamma}_{\theta}(s)=\int_{0}^{s} \frac {\beta(|g(t)|)}{\alpha(|g(t)|)+\theta}\,\mathrm{d}t \quad \mbox{and}\quad \tilde {\gamma}(s)=\int _{0}^{s}\frac{\beta(|g(t)|)}{\alpha(|g(t)|)}\,\mathrm{d}t. \end{aligned}$$
(2.4)

As usual, the usual truncation function \(T_{\theta}\) at level ±θ is defined as \(T_{\theta}(s)=\max\{-\theta, \min\{\theta,s\} \}\). Throughout this paper, we use \(C(\theta_{1},\theta_{2},\ldots,\theta _{m})\) to denote positive constants depending only on specified quantities \(\theta_{1}, \theta_{2},\ldots, \theta_{m}\).

Now we give the definition of weak solutions of problem (\(\mathscr{P}\)).

Definition 2.1

A measurable function \(u\in W_{0}^{1,p}(\Omega)\) is called a weak solution to problem (\(\mathscr {P}\)), if \(a(\cdot, u, \nabla u)\in L^{p'}(\Omega)\) and \(F(\cdot, u, \nabla u)\in L^{1}(\Omega)\) such that
$$ \int_{\Omega}a(x,u,\nabla u)\nabla v\,\mathrm{d}x +\int_{\Omega}F(x,u,\nabla u)v\,\mathrm{d}x =\int _{\Omega}f v\,\mathrm{d}x,\quad \forall v \in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega). $$
(2.5)

For the existence of weak solutions, our result is stated as follows.

Theorem 2.1

If assumptions (H1)-(H5) hold, then there exists at least one bounded weak solution \(u\in L^{\infty}(\Omega)\) to problem (\(\mathscr{P}\)) in the sense of Definition  2.1.

As we have said before, when dealing with the case \(f\in L^{1}(\Omega)\), we shall use the notion of renormalized solution.

Definition 2.2

A measurable function \(u: \Omega\rightarrow \mathbb{R}\) is a renormalized solution of problem (\(\mathscr{P}\)) if
$$\begin{aligned}& T_{k}(u)\in W_{0}^{1,p}(\Omega)\quad \mbox{for any } k\geq0, \end{aligned}$$
(2.6)
$$\begin{aligned}& \lim_{m\rightarrow\infty}\int_{\{m\leq |u|\leq m+1\}}a(x,u, \nabla u)\nabla u\,\mathrm{d}x=0 \end{aligned}$$
(2.7)
and if for any \(h\in W^{1,\infty}(\Omega)\) with compact support and \(\upsilon\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\), u satisfies
$$ \int_{\Omega}a(x,u,\nabla u)\nabla \bigl(h(u) \upsilon\bigr)\,\mathrm{d}x +\int_{\Omega}F(x,u,\nabla u)h(u) \upsilon\,\mathrm{d}x =\int_{\Omega}fh(u)\upsilon\,\mathrm{d}x. $$
(2.8)

The existence result for \(L^{1}\) data is stated as follows.

Theorem 2.2

Assume that (H1) to (H3) hold and \(\frac{\beta}{\alpha}\in L^{1}(\mathbb{R_{+}})\). If \(f\in L^{1}(\Omega)\), then problem (\(\mathscr{P}\)) admits at least one renormalized solution.

Remark 2.2

In Theorem 2.1, the conditions (H4) and (H5) are only needed in proving the \(L^{\infty}(\Omega)\) estimate of u. Therefore in Theorem 2.2, we do not need these assumptions. But instead, we need the condition \(\frac{\beta}{\alpha}\in L^{1}(\mathbb {R_{+}})\) as in [11]. Moreover, by the result of [22], the solution obtained in Theorem 2.2 belongs to \(W_{0}^{1,r}(\Omega)\), provided \(2-\frac{1}{N}< p< N\).

3 Existence of weak solution to problem (\(\mathscr{P}\))

To prove Theorem 2.1, we first establish the \(L^{\infty}\) estimate of solutions to problem (\(\mathscr{P}\)).

Lemma 3.1

Assume that (H1) to (H5) hold. If \(u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) is a weak solution to problem (\(\mathscr{P}\)), then u satisfies the following estimate:
$$ \|u\|_{L^{\infty}(\Omega)}\leq M, $$
(3.1)
where M is a constant which depends only on N, p, q, α, β, \(\|f\|_{L^{q}(\Omega)}\).

Proof of Lemma 3.1

For \(t>0\), \(h>0\), let \(S_{t, h}\) be a real function defined by
$$ S_{t, h}(\eta)= \textstyle\begin{cases} 1,& \eta>t+h , \\ \frac{\eta-t}{h}, & t\leq\eta\leq t+h , \\ 0, & |\eta|\leq t, \\ \frac{\eta+t}{h}, & -t-h\leq\eta\leq-t , \\ -1,& \eta\leq-t-h. \end{cases} $$
(3.2)
It is easy to see that \(S_{t, h}(\phi(u))\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) and so \(S_{t, h}(\phi (u))e^{\gamma_{\theta}(|u|)}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\), where ϕ and \(\gamma _{\theta}\) are defined as in (2.1) and (2.3). Taking \(v=e^{\gamma_{\theta}(|u|)}S_{t, h}(\phi(u))\) as a test function in (2.5), we have
$$\begin{aligned} \begin{aligned} &\frac{1}{h}\int_{\{t< |\phi(u)|\leq t+h\}}\phi'(u)e^{\gamma_{\theta}(|u|)}a(x,u, \nabla u)\nabla u\,\mathrm{d}x \\ &\qquad {}+\int_{\{|\phi(u)|> t\}}\bigl\vert S_{t, h}\bigl( \phi(u)\bigr)\bigr\vert \frac{\beta(|u|)}{\alpha(|u|)+\theta}e^{\gamma_{\theta}(|u|)}a(x,u,\nabla u)\nabla u\,\mathrm{d}x \\ &\qquad {} +\int_{\{|\phi(u)|> t\}}F(x,u,\nabla u)e^{\gamma_{\theta}(|u|)}S_{t, h} \bigl(\phi(u)\bigr)\,\mathrm{d}x \\ &\quad =\int_{\{|\phi(u)|> t\}}fe^{\gamma_{\theta}(|u|)}S_{t, h}\bigl( \phi(u)\bigr)\,\mathrm{d}x. \end{aligned} \end{aligned}$$
Then letting \(\theta\rightarrow0\), we obtain
$$\begin{aligned}& \frac{1}{h}\int_{\{t< |\phi(u)|\leq t+h\}} \phi'(u)e^{\gamma (|u|)}a(x,u,\nabla u)\nabla u\,\mathrm{d}x \\& \qquad {}+\int_{\{|\phi(u)|> t\}}\bigl\vert S_{t, h}\bigl( \phi(u)\bigr)\bigr\vert \frac{\beta(|u|)}{\alpha(|u|)}e^{\gamma(|u|)}a(x,u,\nabla u)\nabla u\,\mathrm{d}x \\& \qquad {} +\int_{\{|\phi(u)|> t\}}F(x,u,\nabla u)e^{\gamma(|u|)}S_{t, h} \bigl(\phi(u)\bigr)\,\mathrm{d}x \\& \quad =\int_{\{|\phi(u)|> t\}}fe^{\gamma(|u|)}S_{t, h}\bigl( \phi(u)\bigr)\,\mathrm{d}x, \end{aligned}$$
(3.3)
where γ is defined as in (2.1). Notice that \(|S_{t, h}(\phi(u))|\leq1\), by (H1), (H3), and applying Hölder’s inequality, we deduce from (3.3) that
$$ \frac{1}{h}\int_{\{t< \omega\leq t+h\}}|\nabla\omega|^{p}\, \mathrm{d}x \leq\int_{\{\omega> t\}}|f|e^{\gamma(|u|)}\,\mathrm{d}x\leq \|f\| _{L^{q}(\Omega)}\biggl(\int_{\{\omega> t\}}\bigl\vert e^{\gamma(|u|)}\bigr\vert ^{q'}\,\mathrm {d}x\biggr)^{\frac{1}{q'}}, $$
where \(\omega=|\phi(u)|=\phi(|u|)\). Let h tend to zero, we find that
$$ -\frac{d}{dt}\int_{\{\omega>t\}}|\nabla \omega|^{p}\,\mathrm{d}x \leq\int_{\{\omega> t\}}|f|e^{\gamma(|u|)} \,\mathrm{d}x\leq\|f\| _{L^{q}(\Omega)}\biggl(\int_{\{\omega> t\}}\bigl\vert e^{\gamma(|u|)}\bigr\vert ^{q'}\,\mathrm {d}x \biggr)^{\frac{1}{q'}}. $$
(3.4)
Setting
$$z(t)=\sup_{\{|s|>\phi^{-1}(t)\}}\frac{e^{\gamma(|s|)}}{(1+\phi (|s|))^{p-1}}, $$
since ϕ is strictly increasing and \(\lim_{s\rightarrow \pm\infty}\phi(s)=0\), we have
$$ \lim_{t\rightarrow +\infty}z(t)=0. $$
(3.5)
Concerning the term \((\int_{\{\omega> t\}}|e^{\gamma(|u|)}|^{q'}\,\mathrm {d}x)^{\frac{1}{q}}\), we have
$$\begin{aligned} \biggl(\int_{\{\omega> t\}}\bigl\vert e^{\gamma(|u|)} \bigr\vert ^{q'}\,\mathrm{d}x\biggr)^{\frac{1}{q}}&= \biggl(\int _{\{\omega> t\}}\biggl(\frac{e^{\gamma(|u|)}}{(1+\omega )^{p-1}}\biggr)^{q'}(1+ \omega)^{q'(p-1)}\,\mathrm{d}x\biggr)^{\frac{1}{q'}} \\ &\leq C(p,q)z(t)\biggl[\biggl(\int_{\{\omega> t\}} \omega^{q'(p-1)}\,\mathrm {d}x\biggr)^{\frac{1}{q'}}+\bigl( \mu_{\omega}(t)\bigr)^{\frac{1}{q'}}\biggr] \\ &\leq C(p,q)z(t)\biggl[\biggl(\int_{0}^{\mu_{\omega}(t)} \omega_{*}^{q'(p-1)}\,\mathrm {d}s\biggr)^{\frac{1}{q'}}+\bigl( \mu_{\omega}(t)\bigr) ^{\frac{1}{q'}}\biggr]. \end{aligned}$$
(3.6)
By (3.4), (3.6), and Lemma 2.1, it follows that
$$\begin{aligned} \begin{aligned}[b] &NC_{N}^{1/N}\mu_{\omega}(t)^{1-1/N} \\ &\quad \leq \bigl(-\mu_{\omega}'(t)\bigr)^{1/p'}\biggl(- \frac{d}{dt}\int_{\{u>t\}}|\nabla \omega|^{p}\, \mathrm{d}x\biggr)^{\frac{1}{p}} \\ &\quad \leq\bigl(-\mu_{\omega}'(t)\bigr)^{1/p'}C(p,q)z^{\frac{1}{p}}(t) \biggl[\biggl(\int_{0}^{\mu _{\omega}(t)}\omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{1}{pq'}}+\bigl(\mu_{\omega}(t) \bigr)^{\frac{1}{pq'}}\biggr], \end{aligned} \end{aligned}$$
(3.7)
which indicates that, for \(0<\theta<\theta+h<|\Omega|\),
$$\begin{aligned} \frac{\omega_{*}(\theta)-\omega_{*}(\theta+h)}{h} \leq&\frac{C(p,q)}{hNC_{N}^{1/N}}\int_{\omega_{*}(\theta+h)}^{\omega _{*}(\theta)}z^{\frac{1}{p}}(t) \frac{(-\mu_{\omega}'(t))^{1/p'}}{\mu_{\omega}(t)^{1-1/N}} \\ &{}\times\biggl[\biggl(\int_{0}^{\mu _{\omega}(t)} \omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{1}{pq'}}+\bigl( \mu_{\omega}(t)\bigr)^{\frac{1}{pq'}}\biggr]\,\mathrm {d}t \\ < &\frac{C(p,q,N)}{h}\sup_{s\in[\omega_{*}(\theta+h),+\infty]}z^{\frac{1}{p}}(s)\int _{\omega_{*}(\theta+h)}^{\omega_{*}(\theta)} \frac{(-\mu_{\omega}'(t))^{1/p'}}{\mu_{\omega}(t)^{1-1/N}} \\ &{}\times\biggl[\biggl(\int _{0}^{\mu _{\omega}(t)}\omega_{*}^{q'(p-1)} \,\mathrm{d}s \biggr)^{\frac{1}{pq'}}+\bigl(\mu_{\omega}(t)\bigr)^{\frac{1}{pq'}}\biggr]\, \mathrm {d}t. \end{aligned}$$
Then we employ (1.15) of [9] to get
$$\begin{aligned} \frac{\omega_{*}(\theta)-\omega_{*}(\theta+h)}{h} < &\frac{C(p,q,N)}{h} \sup_{s\in[\omega_{*}(\theta+h),+\infty]}z^{\frac{1}{p}}(s) \int_{\theta}^{\theta+h} \frac{(-\omega'_{*}(\sigma))^{1/p}}{\sigma^{1-\frac{1}{N}}} \\ &{}\times\biggl[\biggl(\int _{0}^{\sigma}\omega_{*}^{q'(p-1)} \,\mathrm{d}s \biggr)^{\frac{1}{pq'}}+\sigma^{\frac{1}{pq'}}\biggr]\,\mathrm{d}\sigma. \end{aligned}$$
Then letting h tend to zero, we deduce that, for almost \(\theta\in [0,|\Omega|]\),
$$ -\omega'_{*}(\theta) < C(p,q,N)\sup_{s\in[\omega_{*}(\theta),+\infty]}z^{\frac {1}{p}}(s) \frac{(-\omega'_{*}(\theta))^{1/p}}{\theta^{1-\frac{1}{N}}}\biggl[\biggl(\int_{0}^{\theta} \omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{1}{pq'}}+\theta^{\frac{1}{pq'}} \biggr], $$
which leads, after applying Young’s inequality, to
$$\begin{aligned} -\omega'_{*}(\theta) &< C(p,q,N)\Bigl[\sup _{s\in[\omega_{*}(\theta),+\infty]}z^{\frac {1}{p}}(s)\Bigr]^{p'} \frac{1}{\theta^{(1-\frac{1}{N})p'}}\biggl[\biggl(\int_{0}^{\theta} \omega_{*}^{q'(p-1)} \,\mathrm{d}s\biggr)^{\frac{p'}{pq'}}+\theta^{\frac{p'}{pq'}} \biggr] \\ &\leq C(p,q,N)\sup_{s\in[\omega_{*}(\theta),+\infty]}z^{\frac{p'}{p}}(s) \frac{1}{\theta^{(1-\frac{1}{N})p'}}\bigl[\omega_{*}(0)\theta^{\frac {p'}{pq'}}+\theta^{\frac{p'}{pq'}} \bigr]. \end{aligned}$$
(3.8)
Since \(q>\frac{N}{p}\), we have \(q_{0}=\frac{p'}{pq'}+\frac {p'}{N}-p'+1>0\). From (3.5), we deduce that there exists \(t_{0}>0\) such that
$$C(p,q,N)z^{\frac{p'}{p}}(s)|\Omega|^{q_{0}}\leq\frac{1}{2}\quad \mbox{for all }s\geq t_{0}. $$
Hence, upon integration over \([0,\mu_{\omega}(t_{0})]\), inequality (3.8) gives
$$ \omega_{*}(0) \leq1+2t_{0}, $$
which implies that \(\|u\|_{L^{\infty}(\Omega)}\leq\phi^{-1}(1+2t_{0})\). We observe that \(t_{0}\) only depends on p, q, N, \(|\Omega|\), α, β, thus the proof of Lemma 3.1 is finished. □

To prove Theorem 2.1, we shall consider suitable approximate problems. First of all, we recall the following lemma, proved in [12].

Lemma 3.2

There exists a function \(g\in C^{1}(\mathbb{R})\) such that g is odd, strictly increasing, and
$$\begin{aligned}& g'(s)=\alpha\bigl(\bigl\vert g(s)\bigr\vert \bigr)\geq0 \quad \textit{in } \mathbb{R}, \end{aligned}$$
(3.9)
$$\begin{aligned}& g(0)=0,\qquad \lim_{s\rightarrow+\infty}g(s)=+\infty. \end{aligned}$$
(3.10)
For a.e. \(x\in\Omega\), \(\forall s\in\mathbb{R}\), and \(\forall\xi\in \mathbb{R}^{N}\), we define for fixed \(\varepsilon>0\):
$$\begin{aligned}& F_{\varepsilon}(x,s,\xi)=\frac{F(x,s,\xi)}{1+\varepsilon|F(x,s,\xi)|}, \\& a_{\varepsilon}(x,s,\xi)=\varepsilon|\xi|^{p-2}\xi+a \bigl(x,g(s),g'(s)\xi\bigr), \\& a_{\varepsilon l}(x,s,\xi)=\varepsilon|\xi|^{p-2}\xi +a\bigl(x,g \bigl(T_{l}(s)\bigr),g'\bigl(T_{l}(s) \bigr)T'_{l}(s)\xi\bigr). \end{aligned}$$
For any fixed \(\varepsilon>0\), we introduce the approximate problem
$$(\mathscr{P}_{\varepsilon})\quad \left \{ \textstyle\begin{array}{l@{\quad}l} -\operatorname{div}(a_{\varepsilon}(x,u_{\varepsilon},\nabla u_{\varepsilon }))+F_{\varepsilon}(x,g(u_{\varepsilon}),g'(u_{\varepsilon})\nabla u_{\varepsilon})=f_{\varepsilon}&\textit{in } \Omega, \\ u_{\varepsilon}=0& \textit{on } \partial\Omega, \end{array}\displaystyle \right . $$
where \(\{f_{\varepsilon}\}\) satisfy
$$ f_{\varepsilon}\in C_{0}^{\infty}(\Omega)\quad \textit{such that } f_{\varepsilon}\rightarrow f\textit{ strongly in } L^{q}(\Omega) \textit{ as } \varepsilon\rightarrow0. $$

The existence result to problem (\(\mathscr{P}_{\varepsilon}\)) is stated as follows.

Theorem 3.1

Problem (\(\mathscr{P}_{\varepsilon}\)) admits at least a solution \(u_{\varepsilon}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega )\) with \(\|g(u_{\varepsilon})\|_{L^{\infty}(\Omega)}\leq M_{0}\), where \(M_{0}\) is a positive constant depending on M (see Lemma  3.1) and the behavior of function g.

Proof of Theorem 3.1

For any \(l>0\), let us consider the following truncated problem:
$$(\mathscr{P}_{\varepsilon l})\quad \left \{ \textstyle\begin{array}{l@{\quad}l} -\operatorname{div}(a_{\varepsilon l}(x,u_{\varepsilon},\nabla u_{\varepsilon }))+F_{\varepsilon}(x,g(T_{l}(u_{\varepsilon})),g'(T_{l}(u_{\varepsilon }))\nabla T_{l}(u_{\varepsilon}))=f_{\varepsilon}& \mbox{in } \Omega, \\ u_{\varepsilon}=0& \mbox{on } \partial\Omega. \end{array}\displaystyle \right . $$
By the classic result (see [23]), problem (\(\mathscr {P}_{\varepsilon l}\)) admits a solution \(u_{\varepsilon}\in W_{0}^{1,p}(\Omega)\in L^{\infty}(\Omega)\). Then using the same argument of Lemma 3.1, we conclude
$$\bigl\Vert g\bigl(T_{l}(u_{\varepsilon})\bigr)\bigr\Vert _{L^{\infty}(\Omega)}\leq M. $$
In view of Lemma 3.2, it is easy to see that \(g^{-1}\) is defined well and strictly increasing in \(\mathbb{R}\).
Now choosing \(l>g^{-1}(M)\), we obtain
$$ \|u_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq g^{-1}(M). $$
(3.11)
Thus we have \(T_{l}(u_{\varepsilon})=u_{\varepsilon}\), which implies that \(u_{\varepsilon}\) is a weak solution of (\(\mathscr{P}_{\varepsilon}\)). The proof is finished. □

Proof of Theorem 2.1

Taking \(e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}u_{\varepsilon}\) as a test function in problem (\(\mathscr{P}_{\varepsilon}\)), we have
$$\begin{aligned}& \int_{\Omega}e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}a_{\varepsilon}(x,u_{\varepsilon},\nabla u_{\varepsilon})\nabla u_{\varepsilon}\,\mathrm{d}x \\& \qquad {}+\int_{\Omega}|u_{\varepsilon}|\frac{\beta(|g(u_{\varepsilon})|)}{\alpha (|g(u_{\varepsilon})|)+ \theta}e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}a(x,u_{\varepsilon},\nabla u_{\varepsilon})\nabla u_{\varepsilon}\,\mathrm{d}x \\& \qquad {} +\int_{\Omega}F_{\varepsilon}\bigl(x,g(u_{\varepsilon}),g'(u_{\varepsilon}) \nabla u_{\varepsilon}\bigr)e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}u_{\varepsilon}\,\mathrm{d}x \\& \quad =\int_{\Omega}f_{\varepsilon}e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}u_{\varepsilon}\,\mathrm{d}x, \end{aligned}$$
where \(\tilde{\gamma}_{\theta}\) is defined as in (2.4), and g is defined as in Lemma 3.2. Then letting θ tend to zero, using assumptions (H1)-(H4) and Theorem 3.1 we get
$$ \int_{\Omega}e^{\tilde{\gamma}(|u_{\varepsilon}|)}a_{\varepsilon}(x,u_{\varepsilon},\nabla u_{\varepsilon})\nabla u_{\varepsilon}\,\mathrm{d}x \leq\int_{\Omega}f_{\varepsilon}e^{\tilde{\gamma}(|u_{\varepsilon}|)}u_{\varepsilon}\,\mathrm{d}x, $$
where γ̃ is defined as in (2.4).
In view of Theorem 3.1, (H1), and (H2), the above estimate gives
$$ \varepsilon\int_{\Omega}|\nabla u_{\varepsilon}|^{p}+\int_{\Omega}\bigl\vert \nabla g(u_{\varepsilon})\bigr\vert ^{p}\,\mathrm{d}x\leq e^{\tilde{\gamma}(g^{-1}(M))}g^{-1}(M_{0})\|f\|_{L^{1}(\Omega)}. $$
(3.12)
Now denoting \(\bar{u}_{\varepsilon}=g(u_{\varepsilon})\), estimates (3.11) and (3.12) imply that \(\bar{u}_{\varepsilon}\) is bounded uniformly in \(W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). As a consequence, there exist a subsequence (still denoted by \(\{\bar{u}_{\varepsilon}\}\)) and a measurable function \(\bar{u}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) such that
$$\begin{aligned}& \bar{u}_{\varepsilon}\rightharpoonup\bar{u}\quad \mbox{weakly in }W_{0}^{1,p}(\Omega) \mbox{ and weakly}^{\ast}\mbox{ in } L^{\infty }(\Omega), \end{aligned}$$
(3.13)
$$\begin{aligned}& \bar{u}_{\varepsilon}\rightarrow\bar{u} \quad \mbox{a.e. in } \Omega. \end{aligned}$$
(3.14)

In the following, the rest of the proof is divided into several steps.

Step 1: To deal with the difficulty that α vanishes at zero, we define the following truncation function near the origin:
$$ \zeta_{k}(s)=\max\{s,k\}=k+(s-k)_{+},\quad \forall s\in\mathbb{R}, $$
(3.15)
where \(k>0\) is a fixed constant. Then we easily get
$$ \zeta_{k}(\bar{u}_{\varepsilon})\rightharpoonup\zeta_{k}( \bar{u})\quad \mbox{weakly in }W_{0}^{1,p}(\Omega) \mbox{ and weakly}^{\ast}\mbox{ in } L^{\infty}(\Omega). $$
(3.16)
Now taking \(\rho_{\theta}^{\varepsilon}=e^{\gamma_{\theta}(\bar {u}_{\varepsilon})}[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})]_{+}\) as a test function in problem (\(\mathscr {P}_{\varepsilon}\)), by (H1) we have
$$\begin{aligned}& \int_{\Omega}e^{\gamma_{\theta}(\bar{u}_{\varepsilon})}a(x,\bar {u}_{\varepsilon},\nabla \bar{u}_{\varepsilon})\nabla\bigl[ \zeta_{k}(\bar {u}_{\varepsilon}) - \zeta_{k}(\bar{u})\bigr]_{+}\,\mathrm{d}x \\& \qquad {}+\varepsilon\int_{\Omega}e^{\gamma _{\theta}(\bar{u}_{\varepsilon})}|\nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla\bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})\bigr]_{+} \,\mathrm{d}x \\& \qquad {} +\int_{\Omega}\frac{\beta(|\bar{u}_{\varepsilon}|)}{\alpha(|\bar {u}_{\varepsilon}|) +\theta}e^{\gamma_{\theta}(\bar{u}_{\varepsilon})} \bigl[ \zeta_{k}(\bar {u}_{\varepsilon}) -\zeta_{k}(\bar{u}) \bigr]_{+}\alpha\bigl( |\bar{u}_{\varepsilon}|\bigr)|\nabla\bar {u}_{\varepsilon}|^{p} \,\mathrm{d}x \\& \qquad {} +\varepsilon\int_{\Omega}\frac{\beta(|\bar{u}_{\varepsilon}|)}{\alpha (|\bar{u}_{\varepsilon}|) +\theta}e^{\gamma_{\theta}(\bar{u}_{\varepsilon})} \bigl[ \zeta_{k}(\bar {u}_{\varepsilon}) -\zeta_{k}(\bar{u}) \bigr]_{+}|\nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla \bar{u}_{\varepsilon}\,\mathrm{d}x \\& \qquad {} +\int_{\Omega}F_{\varepsilon}(x, \bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon})e^{\gamma_{\theta}(\bar{u}_{\varepsilon})}\bigl[ \zeta_{k}(\bar {u}_{\varepsilon}) -\zeta_{k}(\bar{u})\bigr]_{+} \,\mathrm{d}x \\& \quad \leq\int_{\Omega}f_{\varepsilon}e^{\gamma_{\theta}(\bar{u}_{\varepsilon})}\bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})\bigr]_{+} \,\mathrm{d}x. \end{aligned}$$
(3.17)
It is easy to see that the fourth term of (3.17) is non-negative. So letting θ tend to zero, the above inequality leads to
$$ I_{1}(\varepsilon)+I_{2}(\varepsilon)\leq I_{3}(\varepsilon), $$
(3.18)
where
$$\begin{aligned}& I_{1}(\varepsilon)=\int_{\Omega}e^{\gamma(\bar{u}_{\varepsilon})}a(x, \bar {u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\nabla\bigl[ \zeta_{k}(\bar {u}_{\varepsilon}) -\zeta_{k}(\bar{u})\bigr]_{+} \,\mathrm{d}x, \\& I_{2}(\varepsilon)=\varepsilon\int_{\Omega}e^{\gamma(\bar {u}_{\varepsilon})}| \nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla\bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})\bigr]_{+} \,\mathrm{d}x, \\& I_{3}(\varepsilon)=\int_{\Omega}f_{\varepsilon}e^{\gamma(\bar {u}_{\varepsilon})}\bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) - \zeta_{k}(\bar{u})\bigr]_{+}\,\mathrm{d}x. \end{aligned}$$
Now we estimate all the terms of (3.18).
Estimate of \(I_{2}(\varepsilon)\). Using (3.11), (3.13), and the Hölder inequality, we conclude that
$$\bigl\vert I_{2}(\varepsilon)\bigr\vert \leq\varepsilon e^{\gamma(M_{0})}\biggl(\int_{\Omega} \vert \nabla u_{\varepsilon} \vert ^{p}\,\mathrm{d}x\biggr)^{\frac{p-1}{p}} \biggl[\biggl(\int_{\Omega}\bigl\vert \nabla \zeta_{k}(\bar{u}_{\varepsilon})\bigr\vert ^{p}\,\mathrm {d}x\biggr)^{\frac{1}{p}}+\biggl(\int_{\Omega}\bigl\vert \nabla\zeta_{k}(\bar{u})\bigr\vert ^{p}\,\mathrm {d}x \biggr)^{\frac{1}{p}}\biggr]. $$
Hence, by (3.12) we easily get
$$ \lim_{\varepsilon\rightarrow0}I_{2}(\varepsilon)=0. $$
(3.19)
Estimate of \(I_{3}(\varepsilon)\). By (3.11), (3.14), and the Lebesgue dominated convergence theorem, we infer that
$$ \lim_{\varepsilon\rightarrow0}I_{3}(\varepsilon)=0. $$
(3.20)
Estimate of \(I_{1}(\varepsilon)\). Since \(a(x,s,0)=0\) for a.e. \(x\in\Omega\) and every \(s\in\mathbb{R}\), we obtain
$$\begin{aligned} I_{1}(\varepsilon) =&\int_{\Omega^{k}_{\varepsilon1}}e^{\gamma(\bar {u}_{\varepsilon})}a(x, \bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \cdot\nabla \bigl[ \bar{u}_{\varepsilon}-\zeta_{k}(\bar{u})\bigr]_{+}\,\mathrm{d}x \\ &{}+\int_{\Omega^{k}_{\varepsilon2}} e^{\gamma(\bar{u}_{\varepsilon})}a(x, \bar{u}_{\varepsilon}, \nabla\bar{u}_{\varepsilon}) \cdot\nabla \bigl[ -k -\zeta_{k}(\bar{u}) \bigr]_{+}\,\mathrm{d}x \\ =&\bar{I}_{11}(\varepsilon)+\bar{I}_{12}(\varepsilon), \end{aligned}$$
(3.21)
where
$$ \Omega^{k}_{\varepsilon1}=\{x\in\Omega:\bar{u}_{\varepsilon}< k\}, \qquad \Omega ^{k}_{\varepsilon2}=\{x\in\Omega:\bar{u}_{\varepsilon} \geq k\}. $$
For the term \(\bar{I}_{11}(\varepsilon)\), we can write
$$\begin{aligned} \bar{I}_{11}(\varepsilon) =&\int_{\Omega^{k}_{\varepsilon1}}e^{\gamma(\bar {u}_{\varepsilon})} \bigl[a\bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla \zeta_{k}(\bar{u}_{\varepsilon})\bigr)- a\bigl(x, \zeta_{k}( \bar{u}_{\varepsilon}),\nabla\zeta_{k}(\bar{u})\bigr)\bigr] \cdot \nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}( \bar{u})\bigr]_{+}\,\mathrm{d}x \\ &{}+\int_{\Omega^{k}_{\varepsilon1}}e^{\gamma(\bar{u}_{\varepsilon})} a\bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}(\bar{u}) \bigr) \cdot\nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) - \zeta_{k}(\bar{u})\bigr]_{+}\,\mathrm{d}x. \end{aligned}$$
(3.22)
Collecting (3.11), (3.13), (3.14), and (3.16), it is easy to verify that
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega^{k}_{\varepsilon1}}e^{\gamma(\bar{u}_{\varepsilon})} a\bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}( \bar{u})\bigr) \cdot\nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) - \zeta_{k}(\bar{u})\bigr]_{+}\,\mathrm{d}x=0. $$
(3.23)
Using (3.22), (3.23), (H1), and (H2), we find that
$$\begin{aligned} \varlimsup_{\varepsilon\rightarrow0}\bar{I}_{11}(\varepsilon) \geq& \varlimsup_{\varepsilon\rightarrow0}\int_{\Omega^{k}_{\varepsilon1}}\bigl[a\bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}( \bar{u}_{\varepsilon})\bigr)- a\bigl(x,\zeta_{k}(\bar{u}_{\varepsilon}), \nabla\zeta_{k}(\bar{u})\bigr)\bigr] \\ &{}\cdot\nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})\bigr]_{+} \,\mathrm{d}x \\ =&\varlimsup_{\varepsilon\rightarrow0}\int_{\Omega}\bigl[a\bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}( \bar{u}_{\varepsilon})\bigr)- a\bigl(x, \zeta_{k}( \bar{u}_{\varepsilon}),\nabla\zeta_{k}(\bar{u})\bigr)\bigr] \\ &{}\cdot \nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}( \bar{u})\bigr]_{+}\,\mathrm{d}x, \end{aligned}$$
where we have used the fact \(a(x,s,0)=0\) for a.e. \(x\in\Omega\).
For the term \(\bar{I}_{12}(\varepsilon)\), it is easy to get
$$ \lim_{\varepsilon\rightarrow0}\bar{I}_{12}(\varepsilon)=0. $$
The above two convergence results show that
$$\begin{aligned} \varlimsup_{\varepsilon\rightarrow0}I_{1}(\varepsilon) \geq& \varlimsup_{\varepsilon\rightarrow0}\int_{\Omega}\bigl[a\bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}( \bar{u}_{\varepsilon})\bigr)- a\bigl(x, \zeta_{k}( \bar{u}_{\varepsilon}),\nabla\zeta_{k}(\bar{u})\bigr)\bigr] \\ &{}\cdot \nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}( \bar{u})\bigr]_{+}\,\mathrm{d}x. \end{aligned}$$
(3.24)
Substituting (3.19), (3.20), and (3.24) into (3.18), we conclude
$$\begin{aligned}& \varlimsup_{\varepsilon\rightarrow0}\int_{\Omega}\bigl[a \bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}( \bar{u}_{\varepsilon})\bigr)- a\bigl(x, \zeta_{k}( \bar{u}_{\varepsilon}),\nabla\zeta_{k}(\bar{u})\bigr)\bigr] \\& \quad {}\cdot \nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}( \bar{u})\bigr]_{+}\,\mathrm{d}x\leq0. \end{aligned}$$
(3.25)
Now choosing \(\rho_{\theta}^{\varepsilon}=-e^{\gamma_{\theta}(\bar {u}_{\varepsilon})}[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}(\bar{u})]_{+}\) as a test function in problem (\(\mathscr {P}_{\varepsilon}\)), by the same arguments as in the proof of (3.25) we arrive at
$$\begin{aligned}& \varlimsup_{\varepsilon\rightarrow0}\int_{\Omega}-\bigl[a \bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}( \bar{u}_{\varepsilon})\bigr)- a\bigl(x, \zeta_{k}( \bar{u}_{\varepsilon}),\nabla\zeta_{k}(\bar{u})\bigr)\bigr] \\& \quad {} \cdot \nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}( \bar{u})\bigr]_{-}\,\mathrm{d}x\leq0. \end{aligned}$$
(3.26)
As a consequence of (3.25) and (3.26), we have
$$ \varlimsup_{\varepsilon\rightarrow0}\int_{\Omega}\bigl[a\bigl(x, \zeta_{k}(\bar{u}_{\varepsilon}),\nabla\zeta_{k}( \bar{u}_{\varepsilon})\bigr)- a\bigl(x, \zeta_{k}( \bar{u}_{\varepsilon}),\nabla\zeta_{k}(\bar{u})\bigr)\bigr] \cdot \nabla \bigl[ \zeta_{k}(\bar{u}_{\varepsilon}) -\zeta_{k}( \bar{u})\bigr]\,\mathrm{d}x\leq0. $$
Then, arguing as in [24], we derive that
$$ \nabla\zeta_{k}(\bar{u}_{\varepsilon})\rightarrow\nabla \zeta_{k}(\bar{u})\quad \mbox{strongly in }\bigl(L^{p}( \Omega)\bigr)^{N} \mbox{ and a.e. in } \Omega. $$
(3.27)
Step 2: For any fixed \(k>0\), let us define
$$ \bar{\zeta}_{k}(s)=\min\{s,-k\}=-k+(s+k)_{-},\quad \forall s\in\mathbb {R}. $$
Proceeding as in Step 1, taking \(\rho_{\theta}^{\varepsilon}=e^{\gamma _{\theta}(\bar{u}_{\varepsilon})}[ \bar{\zeta}_{k}(\bar{u}_{\varepsilon}) -\bar{\zeta}_{k}(\bar{u})]_{+}\) and \(\rho_{\theta}^{\varepsilon}=-e^{-\gamma _{\theta}(\bar{u}_{\varepsilon})}[ \bar{\zeta}_{k}(\bar{u}_{\varepsilon}) -\bar{\zeta}_{k}(\bar{u})]_{-}\) as two test functions in problem (\(\mathscr {P}_{\varepsilon}\)), we obtain
$$ \nabla\bar{\zeta}_{k}(\bar{u}_{\varepsilon})\rightarrow\nabla\bar{ \zeta }_{k}(\bar{u}) \quad \mbox{strongly in }\bigl(L^{p}( \Omega)\bigr)^{N} \mbox{ and a.e. in } \Omega. $$
(3.28)
By (3.27) and (3.28), it follows that
$$ \chi_{\{|\bar{u}_{\varepsilon}|\geq k\}}\nabla\bar{u}_{\varepsilon}\rightarrow\chi_{\{|\bar{u}|\geq k\}} \nabla\bar{u}\quad \mbox{strongly in }\bigl(L^{p}(\Omega) \bigr)^{N} \mbox{ and a.e. in } \Omega. $$
(3.29)
In the following, we prove that u is a weak solution to problem (\(\mathscr{P}\)).
Since \(u_{\varepsilon}\) is a weak solution to problem (\(\mathscr{P}\)), it follows that
$$\begin{aligned}& \int_{\Omega}a(x,\bar{u}_{\varepsilon},\nabla \bar{u}_{\varepsilon})\nabla v\,\mathrm{d}x +\varepsilon\int_{\Omega}| \nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla v\, \mathrm{d}x+\int_{\Omega}F_{\varepsilon}(x,\bar{u}_{\varepsilon}, \nabla\bar {u}_{\varepsilon})v\,\mathrm{d}x \\& \quad =\int_{\Omega}f_{\varepsilon}v\,\mathrm{d}x,\quad \forall v\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega). \end{aligned}$$
(3.30)
Concerning the third term on the left-hand side of (3.30), we rewrite it as
$$\begin{aligned}& \int_{\Omega}F(x,\bar{u}_{\varepsilon},\nabla \bar{u}_{\varepsilon}) \upsilon\,\mathrm{d}x \\& \quad = \int_{\{x\in\Omega:|\bar{u}_{\varepsilon}|>k\}} F(x,\bar {u}_{\varepsilon},\nabla \bar{u}_{\varepsilon}) \upsilon\,\mathrm{d}x+ \int_{\{x\in\Omega:|\bar{u}_{\varepsilon}|\leq k\}} F(x, \bar {u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \upsilon\,\mathrm{d}x \\& \quad =I_{1\varepsilon}+I_{2\varepsilon}\quad \mbox{for any fixed } k>0. \end{aligned}$$
(3.31)
To take the limits in \(I_{1\varepsilon}\), we next show that
$$ F(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \chi_{\{|\bar {u}_{\varepsilon}|>k\}}\rightarrow F(x,\bar{u},\nabla\bar{u})\chi_{\{ |\bar{u}|>k\}} \quad \mbox{strongly in } L^{1}(\Omega). $$
(3.32)
Indeed, by (3.14) and (3.29), we already know that \(F(x,t,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\chi_{\{|\bar {u}_{\varepsilon}|>k\}}\rightarrow F(x,t,\bar{u}, \nabla\bar{u})\chi_{\{ |\bar{u}|>k\}}\) almost everywhere in Ω, it suffices to prove the equi-integrability of this sequence and then apply Vitali’s convergence theorem. Using Theorem 3.1 and (H3), we get
$$\bigl\vert F(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \chi_{\{|\bar {u}_{\varepsilon}|>k\}}\bigr\vert \leq C_{0}|\nabla\bar{u}_{\varepsilon}|^{p} \chi_{\{ |\bar{u}_{\varepsilon}|>k\}}, $$
where \(C_{0}\) is a positive constant independent of ε and k. Then the equi-integrability of \(|\nabla\bar{u}_{\varepsilon}|^{p} \chi _{\{|\bar{u}_{\varepsilon}|>k\}}\), which follows from (3.29), indicates that of \(F(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\chi_{\{|\bar{u}_{\varepsilon}|>k\}}\). Therefore, (3.32) is proved.
As a conclusion, we have
$$ \lim_{\varepsilon\rightarrow0}I_{1\varepsilon}={\int_{\{ x\in\Omega: |\bar{u}|> k\}}}F(x, \bar{u},\nabla\bar{u})\upsilon\,\mathrm{d}x, $$
so that
$$ \lim_{k\rightarrow0}\lim_{\varepsilon\rightarrow 0}I_{1\varepsilon}= \int_{\Omega}F(x,\bar{u},\nabla\bar{u}) \upsilon\,\mathrm{d}x. $$
(3.33)
Moreover, by assumption (H3) and (3.12) we get
$$ |I_{2\varepsilon}| \leq\max_{0\leq s\leq k} \beta (s)\int\int _{\{(x,t)\in Q_{\tau}:|\bar{u}_{\varepsilon}(x,t)|\leq k\}} \bigl[|\nabla\bar{u}_{\varepsilon}|^{p}+h(x,t) \bigr]| \upsilon|\,\mathrm{d}x\,\mathrm {d}t \leq C_{1}\max_{0\leq s\leq k} \beta(s), $$
where \(C_{1}\) is a positive constant independent of ε and k. Therefore,
$$ \lim_{k\rightarrow0} \lim_{\varepsilon\rightarrow 0}I_{2\varepsilon}=0, $$
(3.34)
since β is a continuous function from \([0,+\infty)\) into \([0,+\infty)\) and \(\beta(0)=0\).
It follows from (3.31), (3.33), and (3.34) that
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega}F(x,\bar {u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \upsilon\,\mathrm{d}x=\int _{\Omega}F(x,\bar{u},\nabla\bar{u}) \upsilon\,\mathrm{d}x. $$
(3.35)
Similarly, we have
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega}a(x, \bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon})\nabla v\,\mathrm{d}x=\int _{\Omega}a(x,\bar{u},\nabla\bar {u})\nabla v\,\mathrm{d}x. $$
(3.36)
Furthermore, the same argument as (3.19) shows that
$$ \lim_{\varepsilon\rightarrow0} \varepsilon\int_{\Omega}| \nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla v\, \mathrm{d}x=0. $$
(3.37)
Finally, it is easy to see that
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega}f_{\varepsilon}v\,\mathrm{d}x=\int_{\Omega}f v\,\mathrm{d}x. $$
(3.38)
Now letting ε tend to zero, from (3.36)-(3.38), we deduce that ū satisfies (2.5), with u replaced by ū. Thus, the proof is finished. □

4 Existence of renormalized solution to problem (\(\mathscr{P}\))

Proof of Theorem 2.2

By the proof of Theorem 3.1, we deduce that there exists at least one weak solution \(u_{\varepsilon}\) satisfying \(u_{\varepsilon}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) such that
$$\begin{aligned}& \varepsilon\int_{\Omega}|\nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla v\,\mathrm{d}x+\int _{\Omega}a\bigl(x,g(u_{\varepsilon }),\nabla g(u_{\varepsilon})\bigr)\nabla v\,\mathrm{d}x \\& \quad {} + \int_{\Omega}F_{\varepsilon}\bigl(x,g(u_{\varepsilon}), \nabla g(u_{\varepsilon })\bigr)v\,\mathrm{d}x =\int_{\Omega}f_{\varepsilon} v\,\mathrm{d}x,\quad \forall v \in W_{0}^{1,p}(\Omega), \end{aligned}$$
(4.1)
where \(f_{\varepsilon}\) satisfy
$$ f_{\varepsilon}\in C_{0}^{\infty}(\Omega) \quad \mbox{such that } f_{\varepsilon}\rightarrow f\mbox{ strongly in } L^{1}(\Omega) \mbox{ as } \varepsilon\rightarrow0. $$
As before, set \(\bar{u}_{\varepsilon}=g(u_{\varepsilon})\). For any given \(l>s_{0}\) and \(\bar{l}=g^{-1}(l)\), let us take \(v=e^{\tilde{\gamma}_{\theta}(|u_{\varepsilon}|)}T_{\bar{l}}(u_{\varepsilon})\) in (4.1), where \(s_{0}\) is defined as in the proof of Theorem 3.1. Then sending θ tend to zero, using (H1)-(H3) and the fact \(\frac{\beta}{\alpha}\in L^{1}(0,+\infty)\), it follows that
$$ \varepsilon\int_{\Omega}\bigl\vert \nabla T_{\bar{l}}(u_{\varepsilon})\bigr\vert ^{p}\,\mathrm {d}x+ \int_{\Omega}\bigl\vert \nabla T_{l}( \bar{u}_{\varepsilon})\bigr\vert ^{p}\,\mathrm {d}x\leq C, $$
(4.2)
where C is a positive constant independent of ε.
Hence, by the Sobolev space embedding theorem, there exist a measurable function ū and a subsequence (still denoted by \(\{\bar{u}_{\varepsilon}\}\)), such that
$$ \bar{u}_{\varepsilon}\rightarrow\bar{u}\quad \mbox{a.e. in } \Omega $$
(4.3)
and
$$ T_{l}(\bar{u}_{\varepsilon})\rightharpoonup T_{l}(\bar{u}) \quad \mbox{weakly in }W_{0}^{1,p}( \Omega). $$
(4.4)
Step 4.1. In this step, we prove the following result:
$$ \lim_{n\rightarrow\infty} \varlimsup_{\varepsilon\rightarrow 0}{\int _{\{x\in\Omega:n\leq |\bar{u}_{\varepsilon}(x)|\leq n+1\}}}a(x,\bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon})\nabla\bar{u}_{\varepsilon}\,\mathrm{d}x =0. $$
(4.5)
For any integer \(n>1\), define \(\rho_{n}\) by
$$\rho_{n}(r)=T_{n+1}(r)-T_{n}(r),\quad \forall r\in\mathbb{R}. $$
Obviously, we have
$$ 0< |\rho_{n}|\leq1 \quad \mbox{and} \quad \rho_{n}(r)\rightarrow0\quad \mbox{for any } r \mbox{ as }n \rightarrow\infty. $$
(4.6)
Taking \(v=e^{\gamma_{\theta}(|\bar{u}_{\varepsilon}|)}\rho_{n}(\bar {u}_{\varepsilon})\) in (4.1), we get
$$\begin{aligned}& \int_{\Omega} e^{\gamma_{\theta}(|\bar{u}_{\varepsilon}|)}a(x,\bar {u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\nabla\rho_{n}(\bar {u}_{\varepsilon})\,\mathrm{d}x+\int_{\Omega} \rho_{n}(\bar{u}_{\varepsilon})e^{\gamma_{\theta}(|\bar{u}_{\varepsilon}|)}\frac{\beta(|\bar {u}_{\varepsilon}|)}{\alpha(|\bar{u}_{\varepsilon}|)+\theta}a(x, \bar {u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\nabla\bar {u}_{\varepsilon}\,\mathrm{d}x \\& \qquad {} +\int_{\Omega}\varepsilon|\nabla u_{\varepsilon}|^{p-2} \nabla u_{\varepsilon}\nabla\bigl(e^{\gamma_{\theta}(|\bar{u}_{\varepsilon}|)}\rho _{n}( \bar{u}_{\varepsilon})\bigr)\,\mathrm{d}x+\int_{\Omega}F_{\varepsilon}(x, \bar {u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) e^{\gamma_{\theta}(|\bar {u}_{\varepsilon}|)} \rho_{n}(\bar{u}_{\varepsilon})\,\mathrm{d}x \\& \quad = \int_{\Omega} f_{\varepsilon}e^{\gamma_{\theta}(|\bar{u}_{\varepsilon}|)} \rho_{n}(\bar{u}_{\varepsilon})\,\mathrm{d}x. \end{aligned}$$
(4.7)
Passing to the limit as θ tend to zero in (4.7), it follows from (H1) and (H3) that
$$ \int_{\{x\in\Omega:n\leq|\bar{u}_{\varepsilon}(x)|\leq n+1\}} e^{\gamma(|\bar{u}_{\varepsilon}|)}a(x, \bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon})\nabla\bar{u}_{\varepsilon}\,\mathrm{d}x\leq \int_{\Omega} f_{\varepsilon}e^{\gamma(|\bar{u}_{\varepsilon}|)} \rho_{n}(\bar {u}_{\varepsilon})\,\mathrm{d}x. $$
(4.8)
Let \(\varepsilon\rightarrow0\) and then \(n\rightarrow\infty\) in (4.8). Recalling that \(\frac{\beta}{\alpha}\in L^{1}(\mathbb{R}_{+})\), using (4.6) we get
$$ \varlimsup_{\varepsilon\rightarrow0}\int_{\{x\in\Omega :n\leq|\bar{u}_{\varepsilon}(x)|\leq n+1\}} a(x, \bar{u}_{\varepsilon },\nabla\bar{u}_{\varepsilon})\nabla\bar{u}_{\varepsilon}\,\mathrm {d}x\leq \int_{\Omega} f e^{\gamma(|\bar{u}|)} \rho_{n}(\bar{u})\,\mathrm{d}x. $$
(4.9)
It is easy to check that \(\lim_{n\rightarrow\infty}\int_{\Omega} f e^{\gamma(|\bar{u}|)}\rho_{n}(\bar{u})\,\mathrm{d}x=0 \). Thus, passing to the limit as \(n\rightarrow\infty\) in (4.9), the desired result (4.5) follows immediately.
Step 4.2. For any fixed \(k>0\) and \(l>\max\{k,s_{0}\}\), we denote
$$\zeta^{l}_{k}(s)=\max\bigl\{ T_{l}(s),k\bigr\} =k+ \bigl(T_{l}(s)-k\bigr)_{+}, \quad \forall s\in \mathbb{R}. $$
Then we have, in view of (4.3) and (4.4),
$$ \zeta^{l}_{k}(\bar{u}_{\varepsilon}) \rightharpoonup\zeta^{l}_{k}(\bar{u})\quad \mbox{weakly in } W_{0}^{1,p}(\Omega). $$
(4.10)
Let λ be a positive number to be determined, denote
$$\varphi(s)=e^{\lambda s}-1, \quad \forall s\in\mathbb{R} $$
and
$$\rho_{\theta}^{\varepsilon}=e^{\gamma_{\theta}(\bar{u}_{\varepsilon})} \varphi\bigl(\bigl( \zeta^{l}_{k}(\bar{u}_{\varepsilon})-\zeta^{l}_{k}( \bar{u})\bigr)_{+}\bigr) e^{-\gamma_{\theta}(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))}, $$
where \(\gamma_{\theta}\) is defined as in (2.3). We now choose a sequence of increasing function \(S_{n}\in C^{\infty}(\mathbb {R})\) such that
$$ S_{n}(r)=1 \quad \mbox{for }|r|\leq n;\qquad \operatorname{supp}S_{n}\subset[-n-1,n+1]; \qquad \bigl\Vert S'_{n}\bigr\Vert _{L^{\infty}(\mathbb{R})}\leq1. $$
(4.11)
Taking \(v=S_{n}(\bar{u}_{\varepsilon})\rho_{\theta}^{\varepsilon}\) in (4.1), we obtain
$$\begin{aligned}& \hat{I}_{1}(\theta,\varepsilon,n)+\hat{I}_{2}( \theta,\varepsilon,n)+\hat {I}_{3}(\theta,\varepsilon,n) + \hat{I}_{4}(\theta,\varepsilon,n)+\hat{I}_{5}(\theta, \varepsilon,n) \\& \quad \leq\hat{I}_{6}(\theta,\varepsilon,n)+\hat{I}_{7}( \theta,\varepsilon,n) +\hat{I}_{8}(\theta,\varepsilon,n)+ \hat{I}_{9}(\theta,\varepsilon,n), \end{aligned}$$
(4.12)
where
$$\begin{aligned}& \hat{I}_{1}(\theta,\varepsilon,n)=\int_{\Omega} S_{n}(\bar{u}_{\varepsilon })e^{\gamma_{\theta}(\bar{u}_{\varepsilon}) -\gamma_{\theta}(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))} \varphi' \bigl(\bigl(\zeta^{l}_{k}(\bar{u}_{\varepsilon})- \zeta^{l}_{k}(\bar {u})\bigr)_{+}\bigr)a(x,\bar{u}_{\varepsilon}, \nabla\bar{u}_{\varepsilon}) \\& \hphantom{\hat{I}_{1}(\theta,\varepsilon,n)={}}{} \cdot\nabla\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar {u})\bigr)_{+}\bigr) \,\mathrm{d}x, \\& \hat{I}_{2}(\theta,\varepsilon,n)=\varepsilon\int _{\Omega} S_{n}(\bar {u}_{\varepsilon})e^{\gamma_{\theta}(\bar{u}_{\varepsilon}) -\gamma_{\theta}(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))} \varphi'\bigl(\bigl(\zeta ^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr)| \nabla u_{\varepsilon }|^{p-2}\nabla u_{\varepsilon} \\& \hphantom{\hat{I}_{2}(\theta,\varepsilon,n)={}}{} \cdot\nabla\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar {u})\bigr)_{+}\bigr) \,\mathrm{d}x, \\& \hat{I}_{3}(\theta,\varepsilon,n)=\int_{\Omega} S_{n}(\bar{u}_{\varepsilon }) \alpha\bigl(\vert \bar{u}_{\varepsilon} \vert \bigr) \frac{\beta(|\bar{u}_{\varepsilon}|)}{\alpha(|\bar{u}_{\varepsilon }|)+\theta}|\nabla \bar{u}_{\varepsilon}|^{p} \rho_{\theta}^{\varepsilon}\, \mathrm{d}x, \\& \hat{I}_{4}(\theta,\varepsilon,n)=\int_{\Omega} S_{n}'(\bar{u}_{\varepsilon })a(x, \bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\nabla\bar {u}_{\varepsilon} \rho_{\theta}^{\varepsilon}\,\mathrm{d}x, \\& \hat{I}_{5}(\theta,\varepsilon,n)=\varepsilon\int _{\Omega} S_{n}'(\bar {u}_{\varepsilon})|\nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla\bar{u}_{\varepsilon} \rho_{\theta}^{\varepsilon}\,\mathrm{d}x, \\& \hat{I}_{6}(\theta,\varepsilon,n)=\int_{\Omega} S_{n}(\bar{u}_{\varepsilon}) \beta\bigl(\vert \bar{u}_{\varepsilon} \vert \bigr)|\nabla\bar{u}_{\varepsilon}|^{p} \rho _{\theta}^{\varepsilon}\,\mathrm{d}x, \\& \hat{I}_{7}(\theta,\varepsilon,n)=\int_{\Omega} S_{n}(\bar{u}_{\varepsilon }) \frac{\beta(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)}{\alpha(|\zeta ^{l}_{k}(\bar{u}_{\varepsilon})|)+\theta} \varphi\bigl(\bigl( \zeta^{l}_{k}(\bar{u}_{\varepsilon})-\zeta^{l}_{k}( \bar{u})\bigr)_{+}\bigr) a(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \nabla\zeta ^{l}_{k}(\bar{u}_{\varepsilon})\,\mathrm{d}x, \\& \hat{I}_{8}(\theta,\varepsilon,n)=\varepsilon\int _{\Omega} S_{n}(\bar {u}_{\varepsilon}) \frac{\beta(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)}{ \alpha(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)+\theta} \varphi\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr)| \nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla \zeta^{l}_{k}(\bar {u}_{\varepsilon})\,\mathrm{d}x, \\& \hat{I}_{9}(\theta,\varepsilon,n)=\int_{\Omega} S_{n}(\bar{u}_{\varepsilon }) |f_{\varepsilon}| \rho_{\theta}^{\varepsilon}\,\mathrm{d}x. \end{aligned}$$
Limit behaviors of \(\hat{I}_{2}(\theta,\varepsilon,n)\), \(\hat {I}_{5}(\theta,\varepsilon,n)\), and \(\hat{I}_{8}(\theta,\varepsilon,n)\). Thanks to (4.11), we have
$$\begin{aligned} \lim_{\theta\rightarrow0}\hat{I}_{2}(\theta,\varepsilon,n) =& \varepsilon\int_{\Omega} S_{n}'( \bar{u}_{\varepsilon})e^{\gamma(T_{n+1}(\bar{u}_{\varepsilon})) -\gamma(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))} \varphi'\bigl(\bigl( \zeta^{l}_{k}(\bar {u}_{\varepsilon})-\zeta^{l}_{k}( \bar{u})\bigr)_{+}\bigr) \\ &{}\times\bigl\vert \nabla T_{n+1}(u_{\varepsilon})\bigr\vert ^{p-2}\nabla T_{n+1}(u_{\varepsilon})\cdot\nabla\bigl(\bigl( \zeta^{l}_{k}(\bar{u}_{\varepsilon })-\zeta^{l}_{k}( \bar{u})\bigr)_{+}\bigr)\,\mathrm{d}x, \end{aligned}$$
and thus
$$\begin{aligned} \Bigl\vert \lim_{\theta\rightarrow0}\hat{I}_{2}(\theta, \varepsilon,n)\Bigr\vert \leq&\varepsilon C_{1}\int _{\Omega } \bigl\vert \nabla T_{n+1}(u_{\varepsilon}) \bigr\vert ^{p-1}\bigl(\bigl\vert \nabla\zeta^{l}_{k}( \bar {u}_{\varepsilon})\bigr\vert +\bigl\vert \nabla\zeta^{l}_{k}( \bar{u})\bigr\vert \bigr)\,\mathrm{d}x \\ \leq&\varepsilon C_{1}\bigl\Vert \nabla T_{n+1}(u_{\varepsilon}) \bigr\Vert ^{p-1}_{L^{p}(\Omega)}\bigl[\bigl\Vert \nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr\Vert _{L^{p}(\Omega)}+\bigl\Vert \nabla\zeta^{l}_{k}(\bar{u}) \bigr\Vert _{L^{p}(\Omega)}\bigr], \end{aligned}$$
where \(C_{1}\) is a positive constant independent of ε. Therefore, using (4.2) we get
$$ \lim_{\varepsilon\rightarrow0} \lim_{\theta\rightarrow0} \hat{I}_{2}(\theta,\varepsilon,n)=0. $$
(4.13)
Similarly, we have
$$ \lim_{\varepsilon\rightarrow0} \lim_{\theta\rightarrow0} \hat{I}_{5}(\theta,\varepsilon,n)=0 $$
(4.14)
and
$$ \lim_{\varepsilon\rightarrow0} \lim_{\theta\rightarrow0} \hat{I}_{8}(\theta,\varepsilon,n)=0. $$
(4.15)
Limit behaviors of \(\hat{I}_{3}(\theta,\varepsilon,n)\) and \(\hat {I}_{6}(\theta,\varepsilon,n)\). Since
$$\begin{aligned} \hat{I}_{3}(\theta,\varepsilon,n) =&\int_{\{x\in\Omega:\bar{u}_{\varepsilon}(x)\neq0\}} S_{n}'(\bar{u}_{\varepsilon}) \alpha\bigl(\bigl\vert T_{n+1}(\bar {u}_{\varepsilon})\bigr\vert \bigr) \frac{\beta(|T_{n+1}(\bar{u}_{\varepsilon})|)}{\alpha(|T_{n+1}(\bar {u}_{\varepsilon}) |)+\theta} \\ &{}\times\bigl\vert \nabla T_{n+1}( \bar{u}_{\varepsilon})\bigr\vert ^{p} \rho_{\theta}^{\varepsilon}\,\mathrm{d}x, \end{aligned}$$
we get
$$ \lim_{\theta\rightarrow0}\hat{I}_{3}(\theta, \varepsilon,n)= \int_{\Omega} S_{n}'( \bar{u}_{\varepsilon})\varphi\bigl(\bigl(\zeta^{l}_{k}(\bar {u}_{\varepsilon}) -\zeta^{l}_{k}(\bar{u})\bigr)_{+} \bigr)e^{\gamma(\bar{u}_{\varepsilon}) -\gamma(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))}\beta\bigl(\vert \bar{u}_{\varepsilon }\vert \bigr)| \nabla \bar{u}_{\varepsilon}|^{p} \,\mathrm{d}x. $$
(4.16)
As far as \(\hat{I}_{6}(\theta,\varepsilon,n)\) is concerned, we have
$$ \lim_{\theta\rightarrow0}\hat{I}_{6}(\theta, \varepsilon,n)=\int_{\Omega} S_{n}'( \bar {u}_{\varepsilon})\varphi\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})- \zeta^{l}_{k}(\bar{u})\bigr)_{+} \bigr)e^{\gamma(\bar{u}_{\varepsilon}) -\gamma(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))}\beta\bigl(\vert \bar{u}_{\varepsilon }\vert \bigr)| \nabla \bar{u}_{\varepsilon}|^{p} \,\mathrm{d}x. $$
(4.17)
Limit behavior of \(\hat{I}_{4}(\theta,\varepsilon,n)\). From (4.5) and (4.11), it follows that
$$ \lim_{n\rightarrow\infty} \varlimsup_{\varepsilon\rightarrow 0} \lim _{\theta\rightarrow0}\bigl\vert \hat{I}_{4}(\theta, \varepsilon,n)\bigr\vert =0. $$
(4.18)
Limit behavior of \(\hat{I}_{7}(\theta,\varepsilon,n)\). For the term \(\hat{I}_{7}(\theta,\varepsilon,n)\), we have
$$\begin{aligned} \lim_{\theta\rightarrow0}\hat{I}_{7}(\theta, \varepsilon,n)&=\int_{\Omega} S_{n}'( \bar {u}_{\varepsilon})\frac{\beta(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)}{ \alpha(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)} \varphi\bigl(\bigl( \zeta^{l}_{k}(\bar{u}_{\varepsilon})-\zeta^{l}_{k}( \bar{u})\bigr)_{+}\bigr) a(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \nabla\zeta ^{l}_{k}(\bar{u}_{\varepsilon})\,\mathrm{d}x \\ &\leq I_{71}(\varepsilon,n)+I_{72}(\varepsilon,n)+I_{73}( \varepsilon,n), \end{aligned}$$
(4.19)
where
$$\begin{aligned}& I_{71}(\varepsilon,n)=\max_{s\in[k,l]}\frac{\beta(|s|)}{ \alpha(|s|)} \int_{\Omega} \bigl[a\bigl(x,\zeta^{l}_{k}( \bar{u}_{\varepsilon}) ,\nabla\zeta^{l}_{k}( \bar{u}_{\varepsilon})\bigr)-a\bigl(x,\zeta^{l}_{k}(\bar {u}_{\varepsilon}),\nabla\zeta^{l}_{k}(\bar{u})\bigr)\bigr] \\& \hphantom{I_{71}(\varepsilon,n)={}}{}\cdot\nabla\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+} \varphi\bigl(\bigl(\zeta^{l}_{k}(\bar{u}_{\varepsilon})- \zeta^{l}_{k}(\bar {u})\bigr)_{+}\bigr)S_{n}'( \bar{u}_{\varepsilon}) \,\mathrm{d}x, \\& I_{72}(\varepsilon,n)= \int_{\Omega} \frac{\beta(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)}{ \alpha(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)} a\bigl(x,\zeta^{l}_{k}( \bar{u}_{\varepsilon}),\nabla\zeta^{l}_{k}(\bar{u})\bigr) \\& \hphantom{I_{72}(\varepsilon,n)={}}{}\cdot\nabla\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+} \varphi\bigl(\bigl(\zeta^{l}_{k}(\bar{u}_{\varepsilon})- \zeta^{l}_{k}(\bar {u})\bigr)_{+}\bigr)S_{n}'( \bar{u}_{\varepsilon}) \,\mathrm{d}x \end{aligned}$$
and
$$\begin{aligned} I_{73}(\varepsilon,n) =& \int_{\Omega} \frac{\beta(|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)}{\alpha (|\zeta^{l}_{k}(\bar{u}_{\varepsilon})|)} a\bigl(x,\zeta^{l}_{k}( \bar{u}_{\varepsilon}) ,\nabla\zeta^{l}_{k}( \bar{u}_{\varepsilon})\bigr)\nabla\zeta^{l}_{k}(\bar{u}) \\ &{}\times\varphi\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon}) -\zeta^{l}_{k}(\bar{u})\bigr)_{+} \bigr)S_{n}'(\bar{u}_{\varepsilon})\,\mathrm{d}x. \end{aligned}$$
Combining (4.3) with (4.4), we infer that
$$ \lim_{\varepsilon\rightarrow0}I_{72}(\varepsilon,n)=0 $$
(4.20)
and
$$ \lim_{\varepsilon\rightarrow0}I_{73}(\varepsilon,n)=0. $$
(4.21)
Substituting (4.20) and (4.21) into (4.19), we obtain
$$ \varlimsup_{\varepsilon\rightarrow0} \lim_{\theta\rightarrow0} \hat{I}_{7}(\theta,\varepsilon,n)\leq\varlimsup_{\varepsilon \rightarrow0}I_{71}( \varepsilon,n). $$
(4.22)
Limit behavior of \(\hat{I}_{9}(\theta,\varepsilon,n)\). It is straightforward that
$$ \lim_{n\rightarrow\infty} \lim_{\varepsilon\rightarrow0} \lim _{\theta\rightarrow0}\hat{I}_{9}(\theta,\varepsilon,n)=0. $$
(4.23)
Limit behavior of \(\hat{I}_{1}(\theta,\varepsilon,n)\). Note that \(a(x,s,0)=0\) for a.e. \(x\in\Omega\) and every \(s\in\mathbb{R}\), and we get
$$\begin{aligned}& \lim_{\theta\rightarrow0}\hat{I}_{1}(\theta, \varepsilon,n) \\& \quad =\int_{\Omega^{k}_{\varepsilon1}}S_{n}'( \bar{u}_{\varepsilon}) \varphi'\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+} \bigr)a(x, \bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \cdot\nabla \bigl(\zeta^{l}_{k}(\bar{u}_{\varepsilon})- \zeta^{l}_{k}(\bar {u})\bigr)_{+}\,\mathrm{d}x \\& \qquad {} +\int_{\Omega^{k}_{\varepsilon2}} S_{n}'( \bar{u}_{\varepsilon})e^{\gamma (\bar{u}_{\varepsilon}) -\gamma(l)} \varphi'\bigl(\bigl(l- \zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr)a(x,\bar{u}_{\varepsilon}, \nabla\bar {u}_{\varepsilon})\cdot\nabla\bigl(l-\zeta^{l}_{k}( \bar{u})\bigr)_{+}\,\mathrm{d}x \\& \qquad {} +\int_{\Omega^{k}_{\varepsilon3}} S_{n}'( \bar{u}_{\varepsilon})e^{\gamma (\bar{u}_{\varepsilon}) -\gamma(k)} \varphi'\bigl(\bigl(k- \zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr)a(x,\bar{u}_{\varepsilon}, \nabla\bar {u}_{\varepsilon})\cdot\nabla\bigl(k-\zeta^{l}_{k}( \bar{u})\bigr)_{+}\,\mathrm{d}x \\& \quad =\hat{I}_{21}(\varepsilon)+\hat{I}_{22}( \varepsilon)+\hat {I}_{23}(\varepsilon), \end{aligned}$$
(4.24)
where
$$\begin{aligned}& \Omega^{k}_{\varepsilon1}=\{x\in\Omega:k< \bar{u}_{\varepsilon}< l\}, \\& \Omega^{k}_{\varepsilon2}=\{x\in\Omega:\bar{u}_{\varepsilon}\geq l \}, \\& \Omega^{k}_{\varepsilon3}=\{x\in\Omega:\bar{u}_{\varepsilon}\leq k \}. \end{aligned}$$
Using (4.3), (4.4), and (4.11), it is clear that
$$ \lim_{\varepsilon\rightarrow0}\hat{I}_{22}( \varepsilon)=0 $$
(4.25)
and
$$ \lim_{\varepsilon\rightarrow0}\hat{I}_{23}( \varepsilon)=0. $$
(4.26)
Note that \(a(x,s,0)=0\) for a.e. \(x\in\Omega\) and every \(s\in\mathbb{R}\), the term \(\hat {I}_{21}(\varepsilon)\) can be rewritten as follows:
$$\hat{I}_{21}(\varepsilon)=J_{1}(\varepsilon)+J_{2}( \varepsilon), $$
where
$$\begin{aligned}& J_{1}(\varepsilon)=\int_{\Omega} S_{n}'(\bar{u}_{\varepsilon})\bigl[a\bigl(x, \zeta^{l}_{k}(\bar{u}_{\varepsilon}) ,\nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a\bigl(x, \zeta^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \zeta^{l}_{k}(\bar{u})\bigr)\bigr] \\& \hphantom{J_{1}(\varepsilon)={}}{}\cdot\nabla\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr) \varphi'\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr) \,\mathrm{d}x, \\& J_{2}(\varepsilon)=\int_{\Omega} S_{n}'(\bar{u}_{\varepsilon})a\bigl(x, \zeta^{l}_{k}(\bar{u}_{\varepsilon}),\nabla \zeta^{l}_{k}(\bar{u})\bigr) \\& \hphantom{J_{2}(\varepsilon)={}}{}\cdot\nabla\bigl(\bigl( \zeta^{l}_{k}(\bar{u}_{\varepsilon})-\zeta ^{l}_{k}(\bar{u})\bigr)_{+}\bigr)\varphi'\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr) \,\mathrm{d}x. \end{aligned}$$
By (4.3), (4.4), and (4.10), we find that
$$ \lim_{\varepsilon\rightarrow0}J_{2}(\varepsilon)=0. $$
(4.27)
As a direct consequence of (4.24)-(4.27), we have
$$ \varlimsup_{\varepsilon\rightarrow0} \lim_{\theta\rightarrow0} \hat{I}_{1}(\theta,\varepsilon,n)=\varlimsup_{\varepsilon \rightarrow0}J_{1}( \varepsilon). $$
(4.28)
Choosing \(\lambda=2\max_{s\in[k,l]}\frac{\beta(|s|)}{ \alpha(|s|)}\) in the definition of φ, and then combining the limit behaviors of \(\hat{I}_{1}(\theta,\varepsilon,n)\)-\(\hat{I}_{9}(\theta ,\varepsilon,n)\), we get
$$\begin{aligned}& \lim_{n\rightarrow\infty} \varlimsup_{\varepsilon\rightarrow0}\int _{\Omega} S_{n}'(\bar{u}_{\varepsilon}) \bigl[a\bigl(x,\zeta^{l}_{k}(\bar{u}_{\varepsilon}) ,\nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a\bigl(x, \zeta^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \zeta^{l}_{k}(\bar{u})\bigr)\bigr] \\& \quad {}\cdot\nabla\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr) \varphi'\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u})\bigr)_{+}\bigr) \,\mathrm {d}x\leq0, \end{aligned}$$
which yields
$$\begin{aligned}& \lim_{n\rightarrow\infty} \varlimsup_{\varepsilon\rightarrow0}\int _{\Omega} S_{n}'(\bar{u}_{\varepsilon}) \bigl[a\bigl(x,\zeta^{l}_{k}(\bar{u}_{\varepsilon}) ,\nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a\bigl(x, \zeta^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \zeta^{l}_{k}(\bar{u})\bigr)\bigr] \\& \quad {}\cdot\nabla\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar {u})\bigr)_{+}\bigr) \,\mathrm{d}x\leq0. \end{aligned}$$
(4.29)
Step 4.3. Choosing \(v=-S_{n}(\bar{u}_{\varepsilon })e^{-\gamma_{\theta}(\bar{u}_{\varepsilon}) +\gamma_{\theta}(\zeta^{l}_{k}(\bar{u}_{\varepsilon}))} \varphi((\zeta^{l}_{k}(\bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar{u}))_{-}) \) as a test function in (4.1), then arguing as before, we have
$$\begin{aligned}& \lim_{n\rightarrow\infty} \varliminf_{\varepsilon\rightarrow0}\int _{\Omega} S_{n}'(\bar{u}_{\varepsilon}) \bigl[a\bigl(x,\zeta^{l}_{k}(\bar{u}_{\varepsilon}) ,\nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a\bigl(x, \zeta^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \zeta^{l}_{k}(\bar{u})\bigr)\bigr] \\& \quad {}\cdot\nabla\bigl(\bigl(\zeta^{l}_{k}( \bar{u}_{\varepsilon})-\zeta^{l}_{k}(\bar {u})\bigr)_{-}\bigr) \,\mathrm{d}x\geq0. \end{aligned}$$
(4.30)
It follows from (4.29) and (4.30) that
$$\begin{aligned}& \lim_{n\rightarrow\infty} \varlimsup_{\varepsilon\rightarrow0}\int _{\Omega} S_{n}'(\bar{u}_{\varepsilon}) \bigl[a\bigl(x,\zeta^{l}_{k}(\bar{u}_{\varepsilon}) ,\nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a\bigl(x, \zeta^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \zeta^{l}_{k}(\bar{u})\bigr)\bigr] \\& \quad {}\cdot\nabla\bigl(\zeta^{l}_{k}(\bar{u}_{\varepsilon})- \zeta^{l}_{k}(\bar {u})\bigr)\,\mathrm{d}x\leq0. \end{aligned}$$
(4.31)
Taking into account that \(S_{n}'(\bar{u}_{\varepsilon}) a(x,\zeta^{l}_{k}(\bar {u}_{\varepsilon}) ,\nabla\zeta^{l}_{k}(\bar{u}_{\varepsilon}))=a(x,\zeta^{l}_{k}(\bar {u}_{\varepsilon}) ,\nabla\zeta^{l}_{k}(\bar{u}_{\varepsilon}))\) for \(n>l\), using (4.31) we get
$$ \varlimsup_{\varepsilon\rightarrow0}\int_{\Omega} a\bigl(x, \zeta^{l}_{k}(\bar{u}_{\varepsilon}) ,\nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr)\cdot\nabla\bigl( \zeta^{l}_{k}(\bar {u}_{\varepsilon})-\zeta^{l}_{k}( \bar{u})\bigr)\,\mathrm{d}x\leq0, $$
which yields
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega} \bigl[a\bigl(x,\zeta^{l}_{k}(\bar{u}_{\varepsilon}) ,\nabla \zeta^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a\bigl(x, \zeta^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \zeta^{l}_{k}(\bar{u})\bigr)\bigr]\cdot\nabla\bigl(\zeta^{l}_{k}(\bar{u}_{\varepsilon})- \zeta^{l}_{k}(\bar {u})\bigr)\,\mathrm{d}x= 0. $$
(4.32)
Then, arguing as in [24], we derive
$$ \nabla\zeta^{l}_{k}(\bar{u}_{\varepsilon}) \rightarrow\nabla\zeta^{l}_{k}(\bar {u})\quad \mbox{strongly in } \bigl(L^{p}(\Omega)\bigr)^{N} \mbox{ and a.e. in } \Omega. $$
(4.33)
Step 4.4. For any fixed \(l>k>0\), we denote
$$ \bar{\zeta}^{l}_{k}(s)=\min\bigl\{ T_{l}(s),-k\bigr\} =-k-\bigl(T_{l}(s)+k\bigr)_{-}, \quad \forall s\in \mathbb{R}. $$
Choosing \(v=S_{n}(\bar{u}_{\varepsilon})e^{\gamma_{\theta}(\bar {u}_{\varepsilon}) -\gamma_{\theta}(\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon}))} \varphi((\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon})-\bar{\zeta}^{l}_{k}(\bar {u}))_{+})\) as a test function in (4.1), arguing as before we obtain
$$\begin{aligned}& \lim_{n\rightarrow\infty} \varlimsup_{\varepsilon\rightarrow0}\int _{\Omega} S_{n}'(\bar{u}_{\varepsilon}) \bigl[a\bigl(x,\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon}) , \nabla\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a \bigl(x,\bar{\zeta}^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \bar{\zeta}^{l}_{k}(\bar{u})\bigr)\bigr] \\& \quad {}\cdot\nabla\bigl(\bigl(\bar{\zeta}^{l}_{k}( \bar{u}_{\varepsilon})-\bar{\zeta }^{l}_{k}(\bar{u})\bigr)_{+} \bigr)\,\mathrm{d}x\leq0. \end{aligned}$$
Next choosing \(v=-S_{n}(\bar{u}_{\varepsilon})e^{\gamma_{\theta}(\bar{\zeta }^{l}_{k}(\bar{u}_{\varepsilon}))- \gamma_{\theta}(\bar{u}_{\varepsilon})} \varphi((\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon})-\bar{\zeta}^{l}_{k}(\bar {u}))_{-})\) as a test function in (4.1), applying the same argument we get
$$\begin{aligned}& \lim_{n\rightarrow\infty} \varliminf_{\varepsilon\rightarrow0}\int _{\Omega} S_{n}'(\bar{u}_{\varepsilon}) \bigl[a\bigl(x,\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon}) , \nabla\bar{\zeta}^{l}_{k}(\bar{u}_{\varepsilon})\bigr)-a \bigl(x,\bar{\zeta}^{l}_{k}(\bar {u}_{\varepsilon}),\nabla \bar{\zeta}^{l}_{k}(\bar{u})\bigr)\bigr] \\& \quad {}\cdot\nabla\bigl(\bigl(\bar{\zeta}^{l}_{k}( \bar{u}_{\varepsilon})-\bar{\zeta }^{l}_{k}(\bar{u})\bigr)_{-} \bigr) \,\mathrm{d}x\geq0. \end{aligned}$$
Proceeding as in Step 4.3, we infer that
$$ \nabla\bar{\zeta}^{l}_{k}( \bar{u}_{\varepsilon})\rightarrow\nabla\bar {\zeta}^{l}_{k}( \bar{u}) \quad \mbox{strongly in } \bigl(L^{p}(\Omega) \bigr)^{N} \mbox{ and a.e. in } \Omega. $$
(4.34)
As a consequence of (4.33) and (4.34), we have
$$ \chi_{\{|\bar{u}_{\varepsilon}|>k\}}\nabla T_{l}(\bar{u}_{\varepsilon }) \rightarrow\chi_{\{|\bar{u}|>k\}}\nabla T_{l}(\bar{u}) \quad \mbox{strongly in } \bigl(L^{p}(\Omega)\bigr)^{N} \mbox{ and a.e. in } \Omega. $$
(4.35)

Step 4.5. In this step we prove that ū satisfies (2.7), where u is replaced by ū.

For any fixed \(m>k\), one has
$$\begin{aligned}& \int_{\{x\in\Omega:m\leq|\bar{u}_{\varepsilon}(x)|\leq m+1\}} a(x,\bar {u}_{\varepsilon},\nabla \bar{u}_{\varepsilon})\nabla\bar {u}_{\varepsilon}\,\mathrm{d}x \\& \quad = \int_{\Omega} a(x,\bar{u}_{\varepsilon},\nabla \bar{u}_{\varepsilon })\bigl[\nabla T_{m+1}(\bar{u}_{\varepsilon})- \nabla T_{m}(\bar{u}_{\varepsilon})\bigr]\,\mathrm{d}x. \end{aligned}$$
(4.36)
Thus, passing to the limit as ε tends to zero in (4.36), we deduce that, for fixed \(m>k\geq0\),
$$\begin{aligned}& \lim_{\varepsilon\rightarrow0}\int_{\{x\in\Omega:m\leq|\bar{u}_{\varepsilon}(x)|\leq m+1\}} a(x, \bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\nabla\bar {u}_{\varepsilon}\,\mathrm{d}x \\& \quad = \int_{\Omega} a(x,\bar{u},\nabla\bar{u})\bigl[\nabla T_{m+1}(\bar {u})-\nabla T_{m}(\bar{u})\bigr]\,\mathrm{d}x \\& \quad =\int_{\{x\in\Omega:m\leq|\bar{u}|\leq m+1\}} a(x,\bar{u},\nabla\bar {u})\nabla\bar{u}\, \mathrm{d}x. \end{aligned}$$
(4.37)
Taking the limit as m tends to +∞ in (4.37) and using (4.5), we conclude that ū satisfies (2.7).
In the following, we prove that ū satisfies (2.8). Indeed, by (4.1), we have
$$\begin{aligned}& \int_{\Omega}h(\bar{u}_{\varepsilon})a(x, \bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon})\nabla\upsilon\,\mathrm{d}x + \int_{\Omega}\varepsilon h(\bar{u}_{\varepsilon})|\nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla\upsilon\, \mathrm{d}x \\& \qquad {} +\int_{\Omega}h'(\bar{u}_{\varepsilon})a(x, \bar{u}_{\varepsilon},\nabla \bar{u}_{\varepsilon})\nabla\bar{u}_{\varepsilon} \upsilon\,\mathrm{d}x \\& \qquad {}+\int_{\Omega}\varepsilon h'( \bar{u}_{\varepsilon})|\nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla\bar{u}_{\varepsilon} \upsilon\,\mathrm {d}x \\& \qquad {} +\int_{\Omega}h(\bar{u}_{\varepsilon})F_{\varepsilon}(x, \bar {u}_{\varepsilon},\nabla\bar{u}_{\varepsilon})\upsilon\,\mathrm{d}x \\& \quad = \int _{\Omega}h(\bar{u}_{\varepsilon})f_{\varepsilon}\upsilon\, \mathrm{d}x \end{aligned}$$
(4.38)
for any given \(\upsilon\in W^{1,\infty}(\Omega)\) and \(h\in W^{1,\infty }(\mathbb{R})\) such that \(\operatorname{supp}h\subseteq[-l,l]\) for some \(l>0\).
Now we first analyze the fifth term on the left-hand side of (4.38). Recall that \(\operatorname{supp}h\subseteq[-l,l]\), we get
$$h(\bar{u}_{\varepsilon})F(x,\bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon})=h(\bar{u}_{\varepsilon})F\bigl(x,T_{l}( \bar{u}_{\varepsilon}),\nabla T_{l}(\bar{u}_{\varepsilon})\bigr). $$
Therefore, for any k satisfying \(0< k< l\), one has
$$\begin{aligned}& \int_{\Omega}h(\bar{u}_{\varepsilon})F(x, \bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon}) \upsilon\,\mathrm{d}x \\& \quad = \int_{\{x\in\Omega:|\bar{u}_{\varepsilon}|>k\}} h(\bar {u}_{\varepsilon})F \bigl(x,T_{l}(\bar{u}_{\varepsilon}),\nabla T_{l}(\bar {u}_{\varepsilon})\bigr) \upsilon\,\mathrm{d}x \\& \qquad {} + \int_{\{x\in\Omega:|\bar{u}_{\varepsilon}|\leq k\}} h(\bar {u}_{\varepsilon})F \bigl(x,T_{l}(\bar{u}_{\varepsilon}),\nabla T_{l}(\bar {u}_{\varepsilon})\bigr) \upsilon\,\mathrm{d}x \\& \quad =J_{1\varepsilon}+J_{2\varepsilon}. \end{aligned}$$
(4.39)
Similarly to the proof of (3.33) and (3.34), using (4.3) and (4.35) we obtain
$$\begin{aligned} \lim_{k\rightarrow0} \lim_{\varepsilon\rightarrow 0}J_{1\varepsilon}&= \int_{\Omega}h(\bar{u})F\bigl(x,T_{l}(\bar {u}),\nabla T_{l}(\bar{u})\bigr) \upsilon\,\mathrm{d}x \\ &=\int_{\Omega}h(\bar{u})F(x,\bar{u},\nabla\bar{u}) \upsilon\, \mathrm{d}x \end{aligned}$$
(4.40)
and
$$ \lim_{k\rightarrow0} \lim_{\varepsilon\rightarrow 0}J_{2\varepsilon}=0, $$
(4.41)
which imply that
$$ \lim_{\varepsilon\rightarrow 0}\int_{\Omega}h(\bar {u}_{\varepsilon})F(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \upsilon\,\mathrm{d}x=\int_{\Omega}h(\bar{u})F(x,\bar{u},\nabla \bar{u}) \upsilon\,\mathrm{d}x. $$
(4.42)
Similarly, we have
$$ \lim_{\varepsilon\rightarrow 0}\int_{\Omega}h'( \bar {u}_{\varepsilon})a(x,\bar{u}_{\varepsilon},\nabla\bar{u}_{\varepsilon}) \nabla\bar{u}_{\varepsilon}\upsilon\,\mathrm{d}x=\int_{\Omega}h'(\bar {u})a(x,\bar{u},\nabla\bar{u})\nabla\bar{u}\upsilon\, \mathrm{d}x $$
(4.43)
and
$$ \lim_{\varepsilon\rightarrow 0}\int_{\Omega}h(\bar {u}_{\varepsilon})a_{\varepsilon}(x,\bar{u}_{\varepsilon},\nabla\bar {u}_{\varepsilon}) \nabla\upsilon\,\mathrm{d}x=\int_{\Omega}h( \bar {u})a(x,\bar{u},\nabla\bar{u}) \nabla\upsilon\,\mathrm{d}x. $$
(4.44)
As far as the second term of the left-hand side of (4.38) is concerned, by (4.1) we easily get
$$\begin{aligned}& \biggl\vert \int_{\Omega}\varepsilon h(\bar{u}_{\varepsilon})| \nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla\upsilon\, \mathrm{d}x\biggr\vert \\& \quad =\biggl\vert \int_{\Omega}\varepsilon h( \bar{u}_{\varepsilon})\bigl|\nabla T_{\tilde{l}}(u_{\varepsilon})\bigr|^{p-2}\nabla T_{\tilde{l}}(u_{\varepsilon})\nabla\upsilon\, \mathrm {d}x\biggr\vert \\& \quad \leq\varepsilon\sup_{\sigma\in[-l,l]}\bigl\vert h(\sigma)\bigr\vert \bigl\Vert \nabla T_{\tilde{l}}(u_{\varepsilon})\bigr\Vert ^{p-1}_{L^{p}(\Omega)}\|\nabla \upsilon\|_{L^{p}(\Omega)},\quad \mbox{where } \tilde{l}=g^{-1}(l), \end{aligned}$$
thus
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega}\varepsilon h( \bar{u}_{\varepsilon})|\nabla u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}\nabla\upsilon\,\mathrm{d}x=0. $$
(4.45)
Reasoning as in (4.45), one has
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega}\varepsilon h'(\bar{u}_{\varepsilon})|\nabla u_{\varepsilon}|^{p-2} \nabla u_{\varepsilon}\nabla\bar{u}_{\varepsilon} \upsilon\,\mathrm{d}x=0. $$
(4.46)
Finally, it is clear that
$$ \lim_{\varepsilon\rightarrow0}\int_{\Omega}h(\bar {u}_{\varepsilon})f_{\varepsilon}\upsilon\,\mathrm{d}x=\int _{\Omega}h(\bar {u})f\upsilon\,\mathrm{d}x. $$
(4.47)
Then, letting ε tend to zero in (4.38), we conclude from (4.42)-(4.47) that ū satisfies (2.8). Hence, ū is a renormalized solution to problem (\(\mathscr{P}\)). □

Declarations

Acknowledgements

This work is supported by the NSFC of China (No. 11461048), the Natural Science Foundation of Jiangxi Province of China (No. 20132BAB211006), the foundation of Jiangxi Educational Committee (No. GJJ14546).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics and Information Science, Nanchang Hangkong University

References

  1. Fabrie, P, Gallouët, T: Modelling wells in porous media flows. Math. Models Methods Appl. Sci. 10, 673-709 (2000) MATHMathSciNetGoogle Scholar
  2. Gagneux, G, Madaune-Tort, M: Analyse Mathématique de Modèles Non Linéaires de L’ingénierie Pétrolière. Mathématiqes et Applications, vol. 22. Springer, Berlin (1996) Google Scholar
  3. Bertsch, M: A class of degenerate diffusion equations with a singular nonlinear term. Nonlinear Anal. 7, 117-127 (1983) MATHMathSciNetView ArticleGoogle Scholar
  4. Bertsch, M, Peletier, LA: The asymptotic profile of solutions of degenerate diffusion equations. Arch. Ration. Mech. Anal. 91, 207-229 (1985) MathSciNetView ArticleGoogle Scholar
  5. Vázquez, JL: The Porous Medium Equation: Mathematical Theory. Oxford University Press, London (2006) View ArticleGoogle Scholar
  6. Dubinskii, JA: Certain inegalites integrales et resolution de systems équations elliptiques quasilinéaires dégeniries. Mat. Sb. 64(106), 458-480 (1964) MathSciNetGoogle Scholar
  7. Ferone, V, Posteraro, MR, Rakotoson, JM: \(L^{\infty}\)-Estimates for nonlinear elliptic problems with p-growth in the gradient. J. Inequal. Appl. 3(2), 109-125 (1999) MATHMathSciNetGoogle Scholar
  8. Porretta, A, Segura de León, S: Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. 85, 465-492 (2006) MATHMathSciNetView ArticleGoogle Scholar
  9. Rakotoson, JM, Temam, R: Relative rearrangement in quasilinear elliptic variational inequalities. Indiana Univ. Math. J. 36, 757-810 (1987) MathSciNetView ArticleGoogle Scholar
  10. Boccardo, L, Segura de León, S, Trombetti, C: Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term. J. Math. Pures Appl. 80(9), 919-940 (2001) MATHMathSciNetView ArticleGoogle Scholar
  11. Segura de León, S: Existence and uniqueness for \(L^{1}\) data of some elliptic equations with natural growth. Adv. Differ. Equ. 9, 1377-1408 (2003) Google Scholar
  12. Rakotoson, JM: Existence of bounded of some degenerate quasilinear elliptic equations. Commun. Partial Differ. Equ. 12(6), 633-676 (1987) MATHMathSciNetView ArticleGoogle Scholar
  13. Zou, W, Li, F: Existence of solutions for degenerate quasilinear elliptic equations. Nonlinear Anal. 73, 3069-3082 (2010) MATHMathSciNetView ArticleGoogle Scholar
  14. Zou, W, Wang, W: Existence and uniqueness of bounded solutions for a class of degenerate parabolic equations (submitted) Google Scholar
  15. Diperna, RJ, Lions, PL: On the Cauchy problem for the Boltzmann equations: global existence and weak stability. Ann. Math. 130, 321-366 (1989) MATHMathSciNetView ArticleGoogle Scholar
  16. Blanchard, D, Murat, F: Renormalized solutions of nonlinear parabolic problems with \(L^{1}\) data: existence and uniqueness. Proc. R. Soc. Edinb., Sect. A 127(6), 1137-1152 (1997) MATHMathSciNetView ArticleGoogle Scholar
  17. Blanchard, D, Désir, F, Guibé, O: Quasi-linear degenerate elliptic problems with \(L^{1}\) data. Nonlinear Anal. 60, 557-587 (2005) MATHMathSciNetGoogle Scholar
  18. Ammar, K: Renormalized solutions of degenerate elliptic problems. J. Differ. Equ. 234, 1-25 (2007) MATHMathSciNetView ArticleGoogle Scholar
  19. Bénilan, P, Boccardo, L, Gallouët, T, Gariepy, R, Pierre, M, Vázquez, JL: An \(L^{1}\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 22, 240-273 (1995) Google Scholar
  20. Giachetti, D, Maroscia, G: Existence results for a class of porous medium type equations with a quadratic gradient term. J. Evol. Equ. 8, 155-188 (2008) MATHMathSciNetView ArticleGoogle Scholar
  21. Kawohl, B: Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985) MATHGoogle Scholar
  22. Boccardo, L, Gallouët, T: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149-169 (1989) MATHMathSciNetView ArticleGoogle Scholar
  23. Lions, JL: Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires. Dunod, Paris (1969) Google Scholar
  24. Boccardo, L, Murat, F, Puel, JP: Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152(4), 183-196 (1988) MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Zou 2015