\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 32, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/32\hfil Quasilinear elliptic problems]
{Existence of solutions to quasilinear elliptic problems with nonlinearity
and absorption-reaction gradient term}

\author[S. E. Miri \hfil EJDE-2014/32\hfilneg]
{Sofiane El-Hadi Miri}  % in alphabetical order

\address{Sofiane El-Hadi Miri \newline
Universit\'e de Tlemcen, Facult\'e de Technologie, BP 230,
Tlemcen 13000, Alg\'erie. \newline
Laboratoire d'Analyse Non-Lin\'eaire et Math\'ematiques Appliqu\'ees,
 Universit\'e de Tlemcen,
BP 119. Tlemcen, Alg\'erie}
\email{mirisofiane@yahoo.fr}

\thanks{Submitted January 14, 2013. Published Janaury 27, 2014.}
\subjclass[2000]{35D05, 35D10, 35J25, 35J70, 46E30, 46E35}
\keywords{Quasi-linear elliptic problems; entropy solution; general growth}

\begin{abstract}
 In this article we study the  quasilinear elliptic problem
 \begin{gather*}
 -\Delta_p u  =  \pm |\nabla u|^\nu+f(x,u), \quad \text{in } \Omega, \\
 u  \ge  0  \quad  \text{in }\Omega , \\
 u  =  0  \quad \text{on  }\partial\Omega ,
 \end{gather*}
 where $\Omega \subset \mathbb{R}^N$ is a bounded regular domain, $p>1$
 and $0<\nu\le p$. Moreover, $f$ is a nonnegative function verifying suitable
 hypotheses. The main goal of this work is to analyze the interaction between
 the gradient term and the function $f$ to obtain existence results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}\label{section00}

In this article we will discuss existence results for a
class of quasilinear elliptic problems in the form
\begin{equation}
\begin{gathered}
-\Delta _pu=\pm | \nabla u| ^{\nu }+f( x,u)
\quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega ,
\end{gathered}\label{P-}
\end{equation}
where $\Omega \subset \mathbb{R}^N$ is a bounded domain and
$\Delta _pu:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$,
$p>1$, is the classical $p$-Laplace operator and $0<\nu \leq p$.

The function $f:\overline{\Omega }\times [0,+\infty ) \to [0,+\infty )$
is assumed to be   H\"{o}lder continuous, non-decreasing,
 and such that
\begin{gather}
\text{the function }t\mapsto \frac{f( x,t) }{t^{p-1}}\text{ is
non-increasing for all }x\in \overline{\Omega },  \label{H2}
\\
\lim_{t\to 0}\frac{f( x,t) }{t^{p-1}}
=+\infty \text{ and }\lim_{t\to+\infty}\frac{f(x,t) }{t^{p-1}}=0\quad
\text{uniformly for }x\in \overline{\Omega }.
\label{H3}
\\
f( x,0) \neq 0  \label{H4}
\end{gather}
Notice that problems with gradient term are widely studied in the
literature. We can cite the leading works of Boccardo, Gallou\"et, Murat and
their collaborators, see for instance \cite{BGO},\cite{BM} and \cite{BGM}
and the references therein. For some recent works related to our problem, we
can cite \cite{Bou,ADP,APP0,Prim,Rui,AZ,Zou}.

In the particular case $p=2$, problem \eqref{P-} is related to the
Lane-Emden-Fowler and Emden-Fowler equations, treated in many papers; we
particulary cite the works of  Radulescu, and his collaborators
 \cite{GR1,GR2,GR3} and more recently \cite{Dup,K} and the references therein.
For the case without the absence of the gradient term, we refer to \cite{LIN}.

When the nonlinearity is considered as an absorption term we cite \cite{Dal}
where the authors prove the existence of solution even when $\Omega$ is of
infinite measure, and in the same direction we cite \cite{BP}.

The extension to the $p-$laplacian, of the previous results obtained in
the case of the laplacian, especially when using a sub-supersolution method, has
a major difficulty: no general comparison principle for the operator
$-\Delta_p u \pm | \nabla u|^\nu$ exist at our knowledge, and there are only
few partial results in this direction. In addition, the behavior of
the operator changes when considering the cases $p<2$ and $p>2$.
We refer the reader to \cite{Por} for a general discussion about this fact.

\section{Preliminaries}

The next comparison principles will be used frequently in this paper,
for complete proofs of the first three ones we
refer to \cite{Por}  and we refer to \cite{AP} for the last one.

Considering the problem
\begin{equation}
\begin{gathered}
-\operatorname{div}(a(x,\nabla u))+H(x,\nabla u)=f(x) \quad \text{in }\Omega   \\
u=0 \quad \text{on }\partial \Omega
\end{gathered}   \label{Propor}
\end{equation}
and having in mind the particular case
\begin{gather*}
-\Delta _pu \pm| \nabla u| ^{q}=f(x) \quad \text{in  }\Omega  \\
u=0 \quad \text{on }\partial \Omega,
\end{gather*}
with $q\leq p$ we have the following result.

\begin{theorem}[\cite{Por}]
Under the hypotheses: $q>\frac{N(p-1)}{N-1}$, $1<p\leq 2$ and
\begin{gather}
f=f_1(x)+\operatorname{div}(f_2(x))\quad \text{where }
 f_1\in L^{1}(\Omega ), \; f_2\in ( L^{p'}(\Omega )) ^N \label{por1}
\\
[ a(x,\xi )-a( x,\eta ) ] ( \xi -\eta )
\geq \alpha ( | \xi | ^2-| \eta
| ^2) ^{\frac{p-2}{2}}| \xi -\eta |
^2,\quad \alpha >0  \label{por2}
\\
a(x,0)=0  \label{por3}
\\
| a(x,\xi )| \leq \beta ( k(x)+| \xi
| ^{p-1}),\quad \beta >0,\; k(x)\in L^{p'}(\Omega )  \label{por4}
\\
\begin{gathered}
| H(x,\xi )-H( x,\eta ) |  \leq \gamma ( b(x)+| \xi | ^{q-1}+| \eta |
^{q-1}) | \xi -\eta |,
\\
\gamma >0,\quad  b(x)\in L^{r}(\Omega ),
\end{gathered} \label{por5}
\end{gather}
where
\[
1 \leq q\leq p-1+\frac{p}{N}, \quad  r\geq \frac{N(q-( p-1) ) }{q-1}\quad
(\text{with } r=\infty \text{ if }q=1) .
\]
If $u$ and $v$ are respectively sub- and super-solution of \eqref{Propor},
such as
\begin{equation}
( 1+| u| ) ^{\overline{q}-1}u\in W_{0}^{1,p}(\Omega ), \quad
( 1+| v| ) ^{\overline{q}-1}v\in W_{0}^{1,p}(\Omega ),\quad
\overline{q}=\frac{(N-p) ( q-( p-1) ) }{p( p-q)}
\label{por6}
\end{equation}
then $u\leq v$ in $\Omega$.
\end{theorem}

\begin{theorem}[\cite{Por}]
Under the hypotheses: $q<\frac{N(p-1)}{N-1}$, $2-\frac{1}{N}<p\leq 2$,
\eqref{por1}, \ref{por2}, \ref{por3}, \ref{por4}, and
\begin{equation} \label{por7}
\begin{gathered}
| H(x,\xi )-H( x,\eta ) |  \leq \gamma ( b(x)+| \xi | ^{q-1}+| \eta |
^{q-1}) | \xi -\eta | , \\
 \gamma >0,\ b(x)\in L^{r}(\Omega ), \\
r > \frac{N(p-1)}{N(p-1)-( N-1) } , \quad
 1\leq q< \frac{N(p-1)}{( N-1) }.
\end{gathered}
\end{equation}
If $u$ and $v$ are respectively sub- and super-solution of \eqref{Propor},
then $u\leq v$ in $\Omega$.
\end{theorem}

\begin{theorem}[\cite{Por}]
Under the hypotheses: $p>2$, $q>\frac{p}{2}+\frac{(p-1)}{N-1}$,
\eqref{por3}), \eqref{por4}, and
\begin{gather}
[ a(x,\xi )-a( x,\eta ) ] ( \xi -\eta )
\geq \alpha ( 1+| \xi | ^2+| \eta| ^2) ^{\frac{p-2}{2}}| \xi -\eta |^2,\quad
\alpha >0
\\
| H(x,\xi )-H( x,\eta ) |  \leq \gamma ( b(x)+| \xi | ^{q-1}+| \eta |
^{q-1}) | \xi -\eta | ,\quad \gamma >0, \\
b(x) \in L^N(\Omega )\quad \text{where } 1\leq q\leq \frac{p}{2}
+\frac{p}{N}.
\end{gather}
If $u$ and $v$ are respectively sub- and super-solution of \eqref{Propor},
such as
\begin{equation}
( 1+| u| ) ^{\overline{q}-1}u\in W_{0}^{1,p}(\Omega ),\quad
( 1+| v| ) ^{\overline{q}-1}v\in W_{0}^{1,p}(\Omega ),\quad
\overline{q}=\frac{(N-p) ( q-\frac{p}{2}) }{p( \frac{p}{2}+1-q) }
\end{equation}
then $u\leq v$ in $\Omega$.
\end{theorem}


\begin{theorem}[\cite{AP}] \label{compa}
Assume that $1<p$ and let $f$ be a non-negative continuous function such
that $\frac{ f(x,s) }{s^{p-1}}$ is decreasing for $s>0$.
Suppose that $u,v\in W_0^{1,p}(\Omega ) $ are such that
\begin{equation}  \label{eq:u1}
\begin{gathered}
 -\Delta_p u\ge f(x,u), \quad u>0 \text{in  }\Omega,   \\
 -\Delta_pv\le f(x,v), \quad v>0 \text{in  }\Omega.
\end{gathered}
\end{equation}
Then $u\ge v$ in $\Omega $.
\end{theorem}

 Since we are dealing with a generalized notion of solution, we
recall here the definition of entropy solutions for elliptic problems.

\begin{definition} \label{def:entropye0l} \rm
Let $u$ be a measurable function. We say that
$u\in {\mathcal{T} }^{1,p}_{0}(\Omega )$ if $T_k(u)\in W_0^{1,p}(\Omega )$
for all $k>0$, where
\begin{equation} \label{trun}
T_{k}( s) =\begin{cases}
k \operatorname{sgn}( s) &\text{if } | s| \geq k, \\
s  &\text{if } | s| \leq k.
\end{cases}
\end{equation}
Let $H\in L^{1}(\Omega )$. Then $u\in {\mathcal{T} }^{1,p}_{0}(\Omega )$ is an
entropy solution to the problem
\begin{equation}  \label{eq:def}
\begin{gathered}
-\Delta_p u= H \quad \text{in }\Omega , \\
u|_{\partial\Omega }=0,
\end{gathered}
\end{equation}
if for all $k>0$ and all $v\in W_0^{1,p}(\Omega ) \cap L^{\infty}(\Omega )$,
 we have
\begin{equation}  \label{eq:alcala}
\int_\Omega  |\nabla u|^{p-2}\langle\nabla u,\nabla (T_k(u-v))\rangle
=\int_\Omega   H  T_k(u-v).
\end{equation}
\end{definition}

We refer to \cite{BBVP} and \cite{LP} for more properties of entropy solutions.
It is clear that if $u$ is an entropy solution to problem \eqref{P-}, then
$u $ is a distributional solution to \eqref{P-}.

\section{The absorption case}

 In this section we consider the problem
\begin{equation}
\begin{gathered}
-\Delta _pu+| \nabla u| ^{\nu }=f( x,u)  \quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega .
\end{gathered}  \label{abs}
\end{equation}

\begin{theorem} \label{thexi0}
Assume that the assumptions on $f$ hold. If $0<\nu \leq p$,
then problem \eqref{abs} has at least one entropy solution
$u\in W_{0}^{1,p}(\Omega )$ .
\end{theorem}

\begin{proof}
We split the proof into several steps.
\smallskip

\noindent\textbf{Step 1: Construction of supersolution and subsolution.}
To obtain the existence result we will use sub-supersolution argument. Let us
consider the  problem
\begin{equation}
\begin{gathered}
-\Delta _pw=f( x,w)  \quad \text{in }\Omega , \\
w>0 \quad \text{in }\Omega , \\
w=0 \quad \text{on }\partial \Omega .
\end{gathered}   \label{Psup}
\end{equation}
Then under the hypothesis on $f$, problem \eqref{Psup} possesses a unique
solution $w$ which is a supersolution of \eqref{abs}. For the subsolution to
problem \eqref{abs}, we consider $\underline{u}=0$.

Finally by Theorem \ref{compa} we reach that $\underline{u}\leq w$. To obtain
the existence result we use a monotonicity argument. Since no general
comparison principle is known for this kind of problems, we will consider
different values of $p$.

The following steps 2, 3 and 4 are devoted to proving the existence of
solution in the singular case, namely $p<2$, but for different ranges of 
$p$ and $\nu$.
\smallskip

\noindent\textbf{Step 2: Existence result for $\frac{2N}{N+1}\le p<2$ and $
1\le \nu\le p-1+\frac{p}{N}$.}
In this case, by \cite[Theorems 3.1 and 3.2]{Por} we know that a
comparison principle holds for the operator $-\Delta_p u+|\nabla u|^\nu$ in the
space $W_0^{1,p}(\Omega )$.

Then, we define the sequence $\{u_n\}_{n\in \mathbb{N}}$ as follows:
$u_{0}= \underline{u}$ and for $n\geq 1$, $u_n$ is the solution to problem
\begin{equation}
\begin{gathered}
-\Delta _pu_n+| \nabla u_n| ^{\nu }=f(x,u_{n-1})  \quad
 \text{in }\Omega , \\
u_n>0 \quad \text{in }\Omega , \\
u_n=0 \quad \text{on }\partial \Omega .
\end{gathered}   \label{apro1}
\end{equation}
We claim that the sequence $\{u_n\}_{n\in \mathbb{N}}$ is increasing in $n$
and for all $n\geq 0$, $u_n\leq w$. Notice that the last statement follows
easily from Theorem \ref{compa}. To prove the monotonicity of $
\{u_n\}_{n\in \mathbb{N}}$, we will use the comparison result obtained in
\cite{Por}. It is clear that $u_1$ solves
\[
-\Delta _pu_1+| \nabla u_1| ^{\nu }=f(x,u_{0}) .
\]
By the definition of $u_{0}$, we obtain that
\[
-\Delta _pu_1+| \nabla u_1| ^{\nu }
\geq -\Delta _pu_{0}+| \nabla u_{0}| ^{\nu }.
\]
Thus, by the comparison principle in \cite{Por}, we reach $u_1\geq u_{0}$.
Let us show that $u_2\geq u_1$. As above, $u_2$ satisfies
\[
-\Delta _pu_2+| \nabla u_2| ^{\nu }=f(x,u_1) .
\]
Since $f$ is a nondecreasing function, it follows that
\[
-\Delta _pu_2+| \nabla u_2| ^{\nu }\geq -\Delta_pu_1+| \nabla u_1| ^{\nu }.
\]
Hence $u_2\geq u_1$. Therefore, the result follows by induction and then
the claim follows.

 Thus, using $u_n$ as a test function in \eqref{apro1} and by the
non decreasing property of $f$, we obtain that $\|u_n\|_{W_0^{1,p}(\Omega )}\le C$. Hence we obtain the existence of $u\in W_0^{1,p}(\Omega )$ such that $u_n\rightharpoonup u$
weakly in $W_0^{1,p}(\Omega )$ and $u_n\to u$ strongly in $L^\sigma(\Omega )$
for all $\sigma<p^*$.

Since $\underline{u}\le u\le w\in L^\infty(\Omega )$, it follows that
$u\in L^\infty(\Omega )$ and $u_n\to u$ strongly in $L^\sigma(\Omega )$
for all $\sigma\ge 1$.

 Therefore, to have the existence result, we just have to prove that
$| \nabla u_n| ^{\nu }\to | \nabla u|^{\nu }$ in $L^1(\Omega )$.
By the hypothesis on $\nu$, we can see that $\nu<p$, then using $(u-u_n)$ as
a test function in \eqref{apro1}, it follows that
\begin{align*}
&\int_{\Omega } |\nabla u_n|^{p-2}\nabla u_n\nabla
udx-\int_{\Omega } |\nabla u_n|^{p}dx 
+\int_{\Omega } |\nabla u_n|^{\nu }(u-u_n)dx\\
&=\lambda \int_{\Omega } f(x,u_{n-1})(u-u_n)dx.
\end{align*}
By the Dominated Convergence Theorem and as $f$ is assumed to
 be H\"{o}lder continuous, we obtain
\[
\int_{\Omega } f(x,u_{n-1})(u-u_n)dx=o(1).
\]
Now using H\"{o}lder inequality and the fact that $\nu <p$, we obtain
\[
\int_{\Omega } |\nabla u_n|^{\nu }(u-u_n)dx
\leq \Big(\int_{\Omega} |\nabla u_n|^{p}dx\Big)^{\nu/p}
\Big(\int_{\Omega } (u-u_n)^{\frac{p
}{p-\nu }}dx\Big)^{\frac{p-\nu }{p}}=o(1).
\]
We obtain
\[
\int_{\Omega } |\nabla u_n|^{p-2}\nabla u_n\nabla udx-\int_{\Omega
} |\nabla u_n|^{p}dx=o(1).
\]
Then, using Young inequality there results
\begin{align*}
\int_{\Omega }|\nabla u_n|^{p}dx
&=\int_{\Omega }|\nabla u_n|^{p-2}\nabla u_n\nabla udx+o(1)\\
&\leq \frac{p-1}{p}\int_{\Omega }|\nabla u_n|^{p}
 +\frac{1}{p}\int_{\Omega }|\nabla u|^{p}dx+o(1).
\end{align*}
Thus,
\[
\int_{\Omega }|\nabla u_n|^{p}dx\leq \int_{\Omega }|\nabla u|^{p}dx+o(1).
\]
It is clear that
\[
\int_{\Omega }| \nabla u| ^{p}dx
\leq \liminf \int_{\Omega }| \nabla u_n| ^{p}dx
\leq \limsup \int_{\Omega }| \nabla u_n| ^{p}dx
\leq \int_{\Omega }| \nabla u| ^{p}dx.
\]
Therefore, $\|u_n\|_{W_{0}^{1,p}(\Omega )}\to \|u\|_{W_{0}^{1,p}(\Omega )}$
and then $u_n\to u$ strongly in $W_{0}^{1,p}(\Omega )$. Hence the
existence result follows in this case.
\smallskip

\noindent\textbf{Step 3: Existence result for $\frac{2N}{N+1}\le p<2$ and $
p-1+\frac{p}{N}\le \nu\le p$.}
In this case, to get a monotone sequence, we have to change the
approximation. Since $\frac{2N}{N+1}\le p$ then $\nu\ge 1$.

For fixed $n\in \mathbb{N}^{\ast }$, we define the sequence
$\{v_{n,k}\}_{k\in \mathbb{N}}$ as follow: $v_{n,0}=\underline{u}$ and
for $k\geq 1$, $v_{n,k}$ is the solution to problem
\begin{equation}
\begin{gathered}
-\Delta _pv_{k,n}+\frac{| \nabla v_{k,n}| ^{\nu }}{1+
\frac{1}{n}|\nabla v_{k,n}|^{\nu }}=f( x,v_{k-1,n})  \quad \text{in }
\Omega , \\
v_{k,n}>0 \quad \text{in }\Omega , \\
v_{k,n}=0 \quad \text{on }\partial \Omega .
\end{gathered}   \label{apro10}
\end{equation}
Let us begin by proving that the sequence $\{v_{k,n}\}_{k\in \mathbb{N}}$ is
increasing in $k$ and that $v_{k,n}\leq w$, for all $k\geq 0$. For
simplicity, we set
\[
H_n(\xi )=\frac{|\xi |^{\nu }}{1+\frac{1}{n}|\xi |^{\nu }}\quad\text{where  }
\xi \in \mathbb{R}^N.
\]
It is clear that $v_{1,n}$ solves
\[
-\Delta _pv_{1,n}+H_n(\nabla v_{1,n})=f( x,v_{0,n}) .
\]
By the definition of $v_{0,n}$, we obtain that
\[
-\Delta _pv_{1,n}+H_n(\nabla v_{1,n})\geq -\Delta_pv_{0,n}+H_n(\nabla v_{0,n}).
\]
It is clear that $H_n$ satisfies the hypotheses of the comparison
principle in \cite{Por}. Hence we reach $v_{1,n}\geq v_{0,n}$. In the same
way, and using an induction argument, we conclude that $v_{k,n}\geq v_{k-1,n}
$ for all $k\in \mathbb{N}^{\ast }$.

Now, as in the proof of the previous step, using $v_{k,n}$ as a test
function in \eqref{apro10} and by the hypotheses on $f$, we obtain that
$\|v_{k,n}\|_{W_0^{1,p}(\Omega )}\le C$. Thus we obtain the existence of
$u_n\in W_0^{1,p}(\Omega )$ such that $v_{k,n}\rightharpoonup u_n$ weakly
in $W_0^{1,p}(\Omega )$. As in the previous step, we can show that
$v_{k,n}\to u_n$ strongly in $W_0^{1,p}(\Omega )$.

Note that by the previous computation we obtain easily that
\[
v_{k,n}\geq v_{k,n+1}\quad \text{for all }k\geq 1.
\]
Hence we conclude that $u_n$ is the minimal solution to problem
\begin{equation}
\begin{gathered}
-\Delta _pu_n+\dfrac{|\nabla u_n|^{\nu }}{1+\frac{1}{n}|\nabla
u_n|^{\nu }}=f( x,u_n)  \quad \text{in }\Omega , \\
u_n>0 \quad \text{in }\Omega , \\
u_n=0 \quad \text{on }\partial \Omega ,
\end{gathered}   \label{apro100bis}
\end{equation}
with $u_n\leq u_{n+1}$ for all $n\geq 1$. It is clear that
$\underline{u}\leq u_n\leq w\in L^{\infty }(\Omega )$.
Then, as above using $u_n$ as a
test function in \eqref{apro100bis}, we reach that
$\|u_n\|_{W_{0}^{1,p}(\Omega )}\leq C$ and thus, we obtain the
existence of $u\in W_{0}^{1,p}(\Omega )$ such
that $u_n\rightharpoonup u$ weakly in $W_{0}^{1,p}(\Omega )$.

If $\nu<p$, then we follow the above computation to reach that
$u_n\to u$ strongly in $W_0^{1,p}(\Omega )$ and the existence result
holds.

If $\nu =p$, then as in Step 2, we obtain that
\[
f(x,u_{n-1})\to f(x,u)\quad \text{strongly in }L^{1}(\Omega ).
\]
We set $k_n(x)\equiv f(x,u_{n-1})$, then
\[
-\Delta_pu_n+|\nabla u_n|^{p}=k_n(x)
\]
with $k_n\to k\equiv f(x,u)$ strongly in $L^{1}(\Omega )$. Therefore,
using the result of \cite{Por1}, we conclude that $u_n\to u$
strongly in $W_{0}^{1,p}(\Omega )$ and the result follows.
\smallskip

\noindent\textbf{Step 4: Existence result for $\frac{2N}{N+1}\leq p<2$ and $
0<\nu \leq 1$.}
In this case, we adopt a new approximation of the gradient term,
namely we set
\[
Q_n(\xi )=(|\xi |+\frac{1}{n})^{\nu }\quad \text{where  }\xi \in \mathbb{R}^N.
\]
For fixed $n\in \mathbb{N}^{\ast }$, we define the sequence
$\{v_{n,k}\}_{k\in \mathbb{N}}$ as follows: $v_{n,0}=\underline{u}$
and for $k\geq 1$, $v_{n,k}$
is the solution to problem
\begin{equation}
\begin{gathered}
-\Delta _pv_{k,n}+Q_n(\nabla v_{k,n})=f( x,v_{k-1,n})  \quad \text{in }\Omega , \\
v_{k,n}>0 \quad \text{in }\Omega , \\
v_{k,n}=0 \quad \text{on }\partial \Omega .
\end{gathered}  \label{apro100}
\end{equation}
As above we have $v_{k,n}\le w$ for all $k\ge 0$. It is clear that $Q_n$
satisfies the condition of \cite[Theorems 3.1 and 3.2]{Por}.

We claim that the sequence $\{v_{k,n}\}_{k\in \mathbb{N}}$ is increasing in
$k$, for all fixed $n$. To prove the claim, we observe that $v_{1,n}$ solves
\[
-\Delta _pv_{1,n}+Q_n(\nabla v_{1,n})=f( x,v_{0,n}) .
\]
By the definition of $v_{0,n}$, we obtain that
\[
-\Delta _pv_{1,n}+Q_n(\nabla v_{1,n})\geq -\Delta
_pv_{0,n}+Q_n(\nabla v_{0,n}).
\]
Hence, using again the comparison principle in \cite{Por}, we reach that
$v_{1,n}\geq v_{0,n}$. In the same way, using an iteration argument, we
conclude that $v_{k,n}\geq v_{k-1,n}$ for all $k\in \mathbb{N}^{\ast }$ and
then the claim follows.

Now for fixed $k$, we claim that $v_{k,n}\leq v_{k,n+1}$. Using the non
decreasing property and the regularity of $f$ we see that the claim follows
if we can prove that $v_{1,n}\leq v_{1,n+1}$.

By the definition of $v_{1,n}$ and $v_{1,n+1}$, we have
\[
-\Delta _pv_{1,n}+Q_n(\nabla v_{1,n})=-\Delta _pv_{1,n+1}+Q_{n+1}(\nabla
v_{1,n+1})\le -\Delta _pv_{1,n+1}+Q_n(\nabla v_{1,n+1}).
\]
Thus, using the comparison principle of \cite{Por}, we conclude that
$v_{1,n}\le v_{1,n+1}$. The general result follows by induction.

 Now, as in the previous steps, using $v_{k,n}$ as a test function
in \eqref{apro100} and by the H\"{o}lder continuity of $f$, we obtain that
$\|v_{k,n}\|_{W_{0}^{1,p}(\Omega )}\leq C$. Thus, we obtain the existence of
$u_n\in W_{0}^{1,p}(\Omega )$ such that $v_{k,n}\rightharpoonup u_n$ weakly
in $W_{0}^{1,p}(\Omega )$ as $k\to \infty $. The compactness arguments
used in the first step allow us to prove that $v_{k,n}\to u_n$
strongly in $W_{0}^{1,p}(\Omega )$. Hence, we find that $u_n$ is the minimal
solution to problem
\begin{equation}
\begin{gathered}
-\Delta _pu_n+Q_n(\nabla u_n)=f( x,u_n)  \quad \text{in }\Omega , \\
u_n>0 \quad \text{in }\Omega , \\
u_n=0 \quad \text{on }\partial \Omega ,
\end{gathered}  \label{apro1000}
\end{equation}
with $u_n\leq u_{n+1}$ for all $n\geq 1$. It is clear that
$\underline{u} \leq u_n\leq w\in L^{\infty }(\Omega )$.
Then, as above, using $u_n$ as a
test function in \eqref{apro100} we obtain easily that
$\|u_n\|_{W_{0}^{1,p}(\Omega )}\leq C$. Thus, we obtain the existence
of $u\in W_{0}^{1,p}(\Omega )$ such
that $u_n\rightharpoonup u$ weakly in $W_{0}^{1,p}(\Omega )$. Since $\nu <p$,
we conclude that $u_n\to u$ strongly in $W_{0}^{1,p}(\Omega )$ as
above, and the existence result follows.
\smallskip

\noindent\textbf{Step 5: Existence result for $2<p$ and $\nu\le p$.}
To deal with the degenerate case $p>2$, we will make a perturbation in the
principal part of the operator, namely for $\varepsilon>0$, we consider the next
approximating problems
\begin{equation}
\begin{gathered}
-L_{\varepsilon}u+|\nabla u|^{\nu }=f( x,u)  \quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega ,
\end{gathered}  \label{p2}
\end{equation}
where
\[
-L_{\varepsilon}u=-\operatorname{div}((\varepsilon+|\nabla u|^2)^{\frac{p-2}{2}}\nabla u).
\]
We begin by proving that problem \eqref{p2} has a minimal solution $u_{\varepsilon}$
at least for $\varepsilon$ small.
Fixed $\varepsilon>0$, then we define $w_\varepsilon$ to be the unique solution of problem
\begin{equation}
\begin{gathered}
-L_\varepsilon w_\varepsilon= f( x,w_\varepsilon) \quad \text{in }\Omega, \\
w_\varepsilon>0 \quad \text{in }\Omega, \\
w_\varepsilon=0 \quad \text{on }\partial \Omega,
\end{gathered} \label{we}
\end{equation}
(see \cite{M} for the proof of the uniqueness result). It is clear that
$w_\varepsilon$ is a bounded supersolution to \eqref{p2} and $\|w_\varepsilon\|_{L^\infty}\le C$
for all $\varepsilon\ge 0$.
The function $\underline{u}=0$ is aldo a subsolution of \eqref{p2}.

Now, for $\varepsilon$ fixed we define the sequence $\{v_{n,k}\}_{k\in \mathbb{N}}$ as
follows: $v_{n,0}=\underline{u}$ and for $k\geq 1$, $v_{n,k}$ is the
solution to problem
\begin{equation}
\begin{gathered}
-L_{\varepsilon}v_{k,n}+D_n(\nabla v_{k,n})=f( x,v_{k-1,n})  \quad \text{in }\Omega , \\
v_{k,n}>0 \quad \text{in }\Omega , \\
v_{k,n}=0 \quad \text{on }\partial \Omega ,
\end{gathered}   \label{p4}
\end{equation}
where
\[
D_n(\xi )=\begin{cases}
\frac{|\xi |^{\nu }}{1+\frac{1}{n}|\xi |^{\nu }} \quad \text{if  }  1<\nu
\leq p \\
(|\xi |+\frac{1}{n})^{\nu } \quad \text{if  }  \nu \leq 1.
\end{cases}
\]
It is clear that $v_{k,n}\leq w_{\varepsilon}$ for all $k\geq 0$.

We claim that the sequence $\{v_{k,n}\}_{k\in \mathbb{N}}$ is increasing in $k$
for every fixed $n$. To prove the claim, we observe that $v_{1,n}$ solves
\[
-L_{\varepsilon}v_{1,n}+D_n(\nabla v_{1,n})= f( x,v_{0,n}) .
\]
By the definition of $v_{0,n}$, we obtain that
\[
-L_{\varepsilon}v_{1,n}+D_n(\nabla v_{1,n})\geq -L_{\varepsilon}v_{0,n}+D_n(\nabla
v_{0,n}).
\]
Hence, using the comparison principle in \cite[Theorem 4.1]{Por}, we reach
that $v_{1,n}\geq v_{0,n}$. In the same way, using an induction argument, we
conclude that $v_{k,n}\geq v_{k-1,n}$ for all $k\in \mathbb{N}^{\ast }$ and
then the claim follows.

 Using $v_{k,n}$ as a test function in \eqref{p4} we easily get
that $\|v_{k,n}\|_{W_{0}^{1,p}(\Omega )}\leq C$. Thus, we obtain the existence
 of $u_n\in W_{0}^{1,p}(\Omega )$ such that $v_{k,n}\rightharpoonup u_n$ weakly
in $W_{0}^{1,p}(\Omega )$. By the compactness argument used in the Step 2, we
obtain that $v_{k,n}\to u_n$ strongly in $W_{0}^{1,p}(\Omega )$ and $
u_n$ is the minimal solution to the problem
\begin{equation}
\begin{gathered}
-L_{\varepsilon}u_n+D_n(\nabla u_n)=f( x,u_n)  \quad \text{in }\Omega ,\\
u_n>0 \quad \text{in }\Omega , \\
u_n=0 \quad \text{on }\partial \Omega .
\end{gathered}  \label{p5}
\end{equation}
Now, we pass to the limit in $n$.

 Using $u_n$ as a test function in \eqref{p5} and as $f$ is assumed
to be H\"{o}lder continuous, we find that $\|u_n\|_{W_{0}^{1,p}(\Omega )}\leq C$.
Thus, we obtain the existence of $u_{\varepsilon}\in W_{0}^{1,p}(\Omega )$ such that
$u_n\rightharpoonup u_{\varepsilon}$ weakly in $W_{0}^{1,p}(\Omega )$.

 If $\nu <p$, then using the compactness arguments of Step 2 and by
the result of \cite{Por1}, we obtain that $u_n\to u_{\varepsilon}$ strongly
in $W_{0}^{1,p}(\Omega )$. Hence it follows that $u_{\varepsilon}$ is the minimal
solution to problem
\begin{equation}
\begin{gathered}
-L_{\varepsilon}u_{\varepsilon}+|\nabla u_{\varepsilon}|^{\nu }=f( x,u_{\varepsilon})  \quad \text{in }\Omega , \\
u_{\varepsilon}>0 \quad \text{in }\Omega , \\
u_{\varepsilon}=0 \quad \text{on }\partial \Omega .
\end{gathered} \label{p6}
\end{equation}

 If $\nu=p$, then by the argument of the last part of Step 3 and
using the compactness result of \cite{Por1}, we reach the strong convergence
of $\{u_n\}_{n\in \mathbb{N}}$ in $W_0^{1,p}(\Omega )$. Thus, we obtain a minimal
solution to \eqref{p6} also in this case.

To finish, we just have to pass to the limit in $\varepsilon$. Notice that, in
general, the sequence $\{u_{\varepsilon}\}_{\varepsilon}$ is not necessarily monotone in $\varepsilon$.
Using $u_{\varepsilon}$ as a test function in \eqref{p6} we reach that $\|u_{\varepsilon
}\|_{W_{0}^{1,p}(\Omega )}\leq C$ and then $u_{\varepsilon}\rightharpoonup u$ weakly in
$W_{0}^{1,p}(\Omega )$. Since $\underline{u}\leq u_{\varepsilon}\leq w_{\varepsilon}\leq C$, then
we easily get that
\[
f( x,u_{\varepsilon}) \to f( x,u) \text{  strongly in
}L^{1}(\Omega ).
\]
Since $\nu <p$, then using a variation of the compactness result of \cite
{Por1}, there results that $u_{\varepsilon}\to u$ strongly in $W_{0}^{1,p}(\Omega
)$. Hence $u$ solves
\begin{equation}
\begin{gathered}
-\Delta_pu+|\nabla u|^{\nu }=f( x,u)  \quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega ,
\end{gathered}   \label{p61}
\end{equation}
and the existence result follows. It is clear that $\underline{u}\leq u\leq w$.
\end{proof}

\section{The reaction case} \label{sec:reac}

In this section, we study the reaction case, namely we  consider the
problem
\begin{equation}
\begin{gathered}
-\Delta _pu=f( x,u) +| \nabla u| ^{\nu } \quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad  \text{on }\partial \Omega ,
\end{gathered}  \label{reac}
\end{equation}
with $\nu <p-1$.
The main existence result reads as follows.

\begin{theorem} \label{thm4.1}
Suppose that the hypotheses made on $f$ hold. Then, problem \eqref{reac}
 has at least one entropy solution.
\end{theorem}

\begin{proof}
As in the proof of Theorem \ref{thexi0},  problem \eqref{reac} has a
subsolution $\underline{u}=0$. To obtain a supersolution, we first consider
problem
\begin{equation}
\begin{gathered}
-\Delta _pu=f( x,u) +1 \quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega .
\end{gathered} \label{Pint}
\end{equation}
By the assumptions on $f$, we reach that problem \eqref{Pint} has a unique
positive solution $v\in \mathcal{C}^{1,\sigma}(\overline{\Omega })$ with $\sigma<1$.
Then for $C>1$ we have
\[
-\Delta _p( Cv) =C^{p-1}f( x,v) +C^{p-1}.
\]
By hypothesis \eqref{H2}, we obtain $-\Delta _p(Cv)\geq f(x,Cv)+C^{p-1}$.

Since $\nu <p-1$, one can always choose $C$ large enough to have
$C^{p-1}>C^{\nu }|\nabla v|^{\nu }+1$. Thus
\[
-\Delta _p(Cv)\geq f(x,Cv)+|\nabla Cv|^{\nu }+1
\]
and then $\overline{u}=Cv$ is a supersolution to problem \eqref{reac}.

 To prove the existence, we follow the arguments used in the
previous section. By the comparison principle in Theorem \ref{compa} we have
that $\underline{u}\leq \overline{u}$.
\smallskip

\textbf{First case: $\frac{2N}{N+1}\le p<2$ and $\nu<p-1$.}
 Since $p<2$, then $\nu <1$, thus as in the proof of Theorem \ref{thexi0},
we obtain the existence of $u_n$, the minimal solution to problem
\begin{equation}
\begin{gathered}
-\Delta _pu_n=f( x,u_n) +Q_n(\nabla u_n) \quad \text{in }\Omega , \\
u_n>0 \quad \text{in }\Omega , \\
u_n=0 \quad \text{on }\partial \Omega ,
\end{gathered}  \label{reac1}
\end{equation}
where
\[
Q_n(\xi )=(|\xi |+\frac{1}{n})^{\nu },\quad \text{for  }\xi \in \mathbb{R}^N.
\]
It is clear that $\underline{u}\leq u_n\leq \overline{u}$. Using $u_n$
as a test function in \eqref{reac1} and by the fact that $\nu <p-1$, it
follows that $\|u_n\|_{W_{0}^{1,p}(\Omega )}\leq C$.

Then we obtain the existence of $u\in W_0^{1,p}(\Omega )$ such that
$u_n\rightharpoonup u$ weakly in $W_0^{1,p}(\Omega )$.
Notice that $\nu<p$, hence by the previous compactness arguments we
can prove that $u_n\to u$ strongly in $W_0^{1,p}(\Omega )$ and the
existence result follows.
\smallskip

\textbf{Second case: $2<p$ and $\nu<p-1$.}
For fixed $\varepsilon>0$ small, we claim that problem
\begin{equation}
\begin{gathered}
-L_{\varepsilon}u=f( x,u) +|\nabla u|^{\nu } \quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega ,
\end{gathered} \label{reac2}
\end{equation}
where
\[
-L_{\varepsilon}u=-\operatorname{div}((\varepsilon+|\nabla u|^2)^{\frac{p-2}{2}}\nabla u),
\]
has a minimal solution $u_{\varepsilon}$, at leat for $\varepsilon$ small such that
$\underline{u}\leq u_{\varepsilon}\leq \overline{u}$.

 Since $\underline{u},\overline{u}\in \mathcal{C}^{1,\alpha}(\overline{
\Omega })$, then for $\varepsilon$ small we reach that $\underline{u}$ (respectively
$\overline{u}$) is a subsolution (respectively supersolution) to
\eqref{reac2}.

Fix an $\varepsilon$ small enough so that the previous statement still holds true,
and define
\[
D_n(\xi )=\begin{cases}
\frac{|\xi |^{\nu }}{1+\frac{1}{n}|\xi |^{\nu }}& \text{if  }1<\nu <p-1, \\
(|\xi |+\frac{1}{n})^{\nu }&\text{if  }\nu \leq 1.
\end{cases}
\]
Let $u_n$ be the minimal solution to problem
\begin{equation}
\begin{gathered}
-L_{\varepsilon}u_n=f( x,u_n) +D_n(\nabla u_n) \quad \text{in }\Omega ,\\
v_{k,n}>0 \quad \text{in }\Omega , \\
v_{k,n}=0 \quad \text{on }\partial \Omega .
\end{gathered} \label{reac3}
\end{equation}
Notice that $u_n=\lim\limits_{k\to \infty }v_{n,k}$ where the
sequence $\{v_{n,k}\}_{k\in \mathbb{N}}$ is defined as follows:
$v_{n,0}=\underline{u}$ and for $k\geq 1$, $v_{k,n}$ is the solution to problem
\begin{gather*}
-L_{\varepsilon}v_{k,n}=f( x,v_{k-1,n}) +D_n(\nabla v_{k,n}) \quad \text{in }
\Omega , \\
v_{k,n}>0 \quad \text{in }\Omega , \\
v_{k,n}=0 \quad \text{on }\partial \Omega .
\end{gather*}
Using $u_n$ as a test function in \eqref{reac3} and as
$f$ is a nondecreasing H\"{o}lder continuous function, we reach
$\|u_n\|_{W_{0}^{1,p}(\Omega )}\leq C$. Thus, we obtain the existence
of $u_{\varepsilon}\in W_{0}^{1,p}(\Omega )$ such that $u_n\rightharpoonup u_{\varepsilon}$
weakly in $W_{0}^{1,p}(\Omega )$. By the compactness argument in Step 2 of
Theorem \ref{thexi0} we obtain that $u_n\rightharpoonup u_{\varepsilon}$ strongly
in $W_{0}^{1,p}(\Omega )$ and $u_{\varepsilon}$ is the minimal solution to \eqref{reac2}.
It is clear that $\underline{u}\leq u_{\varepsilon}\leq \overline{u}$, and the claim
follows.

The last step is to pass to the limit in $\varepsilon$. Using $u_\varepsilon$ as a test
function in \eqref{reac2}, we reach that $\|u_\varepsilon\|_{W_0^{1,p}(\Omega )}\le C$
and then $u_\varepsilon\to u$ weakly in $W_0^{1,p}(\Omega )$.

Since $\nu <p$, a modification of the arguments used in the proof of Theorem
\ref{thexi0}, allows us to obtain that $u_{\varepsilon}\to u$ strongly in
 $W_{0}^{1,p}(\Omega )$. Thus $u$ solves
\begin{equation}
\begin{gathered}
-\Delta_pu=f( x,u) +|\nabla u|^{\nu } \quad \text{in }\Omega , \\
u>0 \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega .
\end{gathered}\label{p62}
\end{equation}
\end{proof}

\begin{remark} \rm
Observe that the condition \ref{H4} imposed on $f$ to ensure that $0$
is a strict subsolution, is not necessary, indeed one can drope it, and
consider as subsolution the function introduced in \cite{Dup}, in \cite{M}
and in \cite{M2}, defined by $\ ~\underline{u}$ $=Mh( c\varphi_1)$ where $M$
and $c$ are positive constants to be chosen, $\varphi _1$ is the first
eigenfunction of the p-laplacian and $h$ is the solution to the differential
equation
\begin{gather*}
h''(t)=q(h(t))g(h(t)), \\
h>0, \quad h'>0, \\
h(0)=h'(0)=0.
\end{gather*}
where $q:( 0,+\infty ) \to ( 0,+\infty ) $ is a non-increasing and H\"older
continuous function, and $g(s)$ behaves like $\frac{1}{s^\beta}$,
for some $\beta>0$.
\end{remark}

\subsection*{Acknowledgments}
I am deeply grateful to Professors B. Abdellaoui and V. Radulescu,
and to the anonymous referees for providing constructive comments
that help in improving the contents of this article.

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\end{document}