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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 204, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/204\hfil The $(n-1)$-radial symmetric solution]
{The $(n-1)$-radial symmetric positive classical solution
 for elliptic equations with gradient}

\author[Y. Zhang, Q. Xu, P. Zhao \hfil EJDE-2013/204\hfilneg]
{Yong Zhang, Qiang Xu, Peihao Zhao}  

\address{Yong Zhang \newline
Department of Mathematics, Lanzhou University,
Lanzhou, Gansu, 730000,  China \newline
Department of Mathematics, Chizhou College,
Chizhou, Anhui, 247000, China}
\email{zhangy12@lzu.edu.cn}

\address{Qiang Xu \newline
Department of Mathematics, Lanzhou University,
Lanzhou, Gansu, 730000,  China}
\email{xuqiang09@lzu.edu.cn}

\address{Peihao Zhao \newline
Department of Mathematics, Lanzhou University,
Lanzhou, Gansu, 730000,  China}
\email{zhaoph@lzu.edu.cn}


\thanks{Submitted June 14, 2013. Published Spetember 16, 2013.}
\subjclass[2000]{35J60, 35B09}
\keywords{Elliptic equations; symmetric;  positive
 solution; a priori estimates; \hfill\break\indent
 fixed point theorem}

\begin{abstract}
 In this article, we study the existence of the $(n-1)$-radial
 symmetric positive classical solution for elliptic equations with
 gradient. By some special techniques in two variables, we show
 a priori estimates, and then show the existence of a solution using
 a fixed point theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider the following boundary-value problem of a 
second-order  elliptic equation,
\begin{equation}
\begin{gathered}
-\Delta u=f(x,u,\nabla u)\quad  \text{in } \Omega ,\\
 u(x)=0 , \quad \text{on } \partial\Omega,
\end{gathered}\label{e1.1}
\end{equation}
where  $\Omega$ is a bounded  domain in  $\mathbb{R}^n$,
$n\geq 3$.

This type of equations have  been  studied  by several authors. As
the nonlinearity $f$ depends on the gradient of the solution,
solving \eqref{e1.1} is not variational and
 the well developed critical point theory can not be applied
directly. But if  $f$ has a special form, by changing variables,
\eqref{e1.1} can be transformed into a boundary-value problem which is
independent of $\nabla u$. For example, When
$f(x,u,\nabla u)=g(u)+\lambda|\nabla u|^2+\eta$,
Ghergu and R\u{a}dulescu \cite{Ghergu1}  used the above method to show the
existence of  positive classical solution under the assumption that
$g$ is decreasing and unbounded at the origin.
A similar method appears in \cite{Abdellaoui}, where
$f(x,u,\nabla u)$ has critical growth with respect to $\nabla u$;
see also \cite{Ghergu2,Zou}.
In addition,  Chen and Yang \cite{Chen Wenjing} considered the existence of
positive solutions for \eqref{e1.1} on a smooth compact Riemannian manifold.
 As far as we know, the methods used to
solve \eqref{e1.1} are mainly sub and super-solution, fixed point theorems,
Galerkin method, and topological degree, see, for instance,
\cite{Alves, Amann,Figueiredo2,Pohozaev,Wang,Xavier,Yan}.

It is worth mentioning that  de Figueiredo, Girardi and Matzeu \cite{Figueiredo1}
developed a quite different method of variational type.
Firstly, for each $\omega\in H_0^1(\Omega)$,
they considered the  boundary problem
\begin{equation}
\begin{gathered}
-\Delta u=f(x,u,\nabla \omega)\quad  \text{in } \Omega ,\\
u(x)=0 , \quad \text{on } \partial\Omega.
\end{gathered}\label{e1.2}
\end{equation}
which is a variational problem. 
Under the assumptions that $f$ has
a superlinear subcritical growth at zero and at infinity with
respect to the second variable, and $f$ is locally Lipschitz
continuous with the third variable, they proved that a weak
solution $u_{\omega}$ of \eqref{e1.2} exists by mountain-pass theorem. Then
they have constructed a sequence $\{u_k\}\subset H_0^1(\Omega)$ as solutions of
\begin{equation}
\begin{gathered}
-\Delta u_n=f(x,u_n,\nabla u_{n-1})\quad  \text{in } \Omega ,\\
u_n(x)=0 , \quad \text{on } \partial\Omega,
\end{gathered}\label{e1.3}
\end{equation}
and verified that $\{u_k\}$ converges to  a solution of
\eqref{e1.1}. However, this solution is just in $H_0^1(\Omega)$.

Additionally, the existence of classical solutions for \eqref{e1.1} has
been  obtained by mountain-pass lemma and a suitable truncation
method in \cite{Girardi}, but the conditions imposed on $f$ are very
strong:
\begin{enumerate}
\item  $f$ is locally Lipschitz continuous on
$\bar{\Omega}\times \mathbb{R}\times \mathbb{R}^n$,

\item  $\frac{f(x,t,\xi)}{t}$ converges to zero uniformly
with respect to  $x\in \Omega$, $\xi\in \mathbb{R}^n $ as $t$ tends to zero,

\item  there exist $a_{1}>0$, $p\in (1,\frac{n+2}{n-2})$ and
$r\in (0,1)$ such that
\[
|f(x,t,\xi)|\leq a_{1}(1+|t|^{p})(1+|\xi|^{r}),\quad \forall x\in
\bar{\Omega},\; t\in \mathbb{R},\; \xi\in \mathbb{R}^n,
\]

\item  there exist $\vartheta >2$ and $a_{2}, a_{3}, t_0>0$ such that
\begin{gather*}
0<\vartheta F(x,t,\xi)\leq tf(x,t,\xi),\quad \forall x\in
\bar{\Omega},\; t\geq t_0,\; \xi\in \mathbb{R}^n,\;
F(x,t,\xi)\geq a_{2}|t|^{\vartheta}-a_{3}; \\
F(x,t,\xi)\geq a_{2}|t|^{\vartheta}-a_{3},
\end{gather*}
where $F(x,t,\xi)=\int_0^{t}f(x,s,\xi)ds$.
\end{enumerate}

 As far as we know, a few authors have paid attention to 
the radial solutions of \eqref{e1.1}; see for example 
\cite{Bisci, Figueiredo2}. 
So we will limit us to the radially symmetric case and
try to focus on some new methods to  study \eqref{e1.1}. 
We consider the  boundary-value problem \eqref{e1.1} and assume 
the following: 
\begin{itemize}
\item[(D1)] $\Omega$ is a so-called $(n-1)$-symmetric domain
in $\mathbb{R}^n$($n\geq 3$), that is, $\Omega$ is symmetric with
respect to $x_{1},x_{2},\cdots,x_{n-1}$ and
$0\notin\overline{\Omega}$;

\item[(F1)]  $f(x,u,\eta)$ is a nonnegative function satisfying 
$f(x,u,\eta)=f(r,x_n,u,|\eta|)$, where
$r=\sqrt{x_{1}^2+x_{2}^2+\cdots+x_{n-1}^2}$;


\item[(F2)]  there exist $c_0\geq 1$, $M>0$, $ p>1$,
$\tau\in(0,\frac{2p}{p+1}) $ such that
\[
u^{p}-M|\eta|^{\tau}\leq  f(x,u,\eta)\leq
c_0u^{p}+M|\eta|^{\tau},\quad \forall
(x,u,\eta)\in\Omega\times\mathbb{R}\times\mathbb{R}^n;
\]

\item[(F3)] $f(x,u,\eta)\in C^{\beta}(\Omega,\mathbb{R},\mathbb{R}^n)$ for some
$\beta\in(0,1)$.

\end{itemize}

We remark that in \cite{Ruiz}, the constants $p$ and $\tau$
belong to $(1 , \frac{2(n-1)}{n-2})$ and $(1,\frac{2p}{p+1})$
respectively. Obviously,  the conditions  in (F2) are
weaker than those in \cite{Ruiz}.


If the solution $u(x)$ is $(n-1)$-radial symmetric, that is
$u(x)=u(r,x_n)$, then by (F1) Equation \eqref{e1.1} can be transformed
into the following elliptic equation in two variables:
\begin{equation}
\begin{gathered}
-(u_{rr}+u_{x_nx_n})=H(r,x_n,u,u_{r},u_{x_n}),\quad  \text{in } \Omega ,\\
u(x)=0 , \quad \text{on } \partial\Omega,
\end{gathered}\label{e1.4}
\end{equation}
where $H(r,x_n,u,u_{r},u_{x_n})=f(r,x_n,u,|\nabla u|)+\frac{n-2}{r}u_{r}$.
 Motivated by the priori estimates mentioned in  \cite{Ruiz} and special 
technique for the equation in two variables developed in
 \cite{Gilbarg},  we develop an approach  which is distinct from the previous  
works, and  shows the existence of the  $(n-1)$-radial symmetric positive 
classical $C^{2,\beta}$-solutions of  \eqref{e1.1}.
Note that solution in \cite{Ruiz} is just in $C^{1,\alpha}(\Omega)$.


 The rest of this  work is organized as follows.
 Motivated by \cite{Ruiz} we give a priori estimates in section 2. 
In section 3 we show the existence of  $(n-1)$-radial symmetric positive 
classical solutions with the help of \cite{Gilbarg}.

\section{A priori estimates}

Compared with the reference \cite{Ruiz}, we should deal with  the
second term $\frac{n-2}{r}u_{r}$ of $H(r,x_n,u,u_{r},u_{x_n})$
in \eqref{e1.4} additionally, it is necessary  to give a brief proof of
the a priori estimates although the process is similar to that in
\cite{Ruiz}.

\begin{theorem} \label{thm2.1}
Assume that {\rm (D1)} and {\rm (F2)}  hold, and that $\lambda<\lambda_0$ for
some $\lambda_0$ fixed. Then, for any $C^1$-solution $u$ of the
equation
\begin{equation}
\begin{gathered}
-(u_{rr}+u_{x_nx_n})=H(r,x_n,u,u_{r},u_{x_n})+\lambda, \quad  \text{in } \Omega ,\\
u(x)=0, \quad \text{on } \partial\Omega,
\end{gathered}\label{e2.1}
\end{equation}
there exists a positive constant $C$ such that 
$\sup_{\Omega} u <C$.
\end{theorem}

To prove this theorem, we need the following lemmas.

\begin{lemma} \label{lem2.1}
Let {\rm (D1)} hold and $u(r,x_n)$ be a positive weak $C^1$-solution
of the inequality
\begin{equation}
-(u_{rr}+u_{x_nx_n})\geq u^{p}-M|\nabla
u|^{\tau}+\frac{n-2}{r}u_{r},\label{e2.2}
\end{equation}
 where $1<p$ and $0<\tau<2p/(p+1)$.  Take $\gamma\in (0,p)$
and $\mu\in (0,\frac{2p}{p+1})$. Denote by $B_{2R}$ a ball of radius
$2R$ contained in $\Omega$, where $R<R_0$ and $R_0$ is a
positive constant. Then there exists a  positive constant 
$C=C(p,\gamma, \mu, R_0)$ such that
\begin{gather}
\int_{B_{R}}u^{\gamma}\leq CR^{2-2\gamma/(p-1)},\label{e2.3} \\
\int_{B_{R}}|\nabla u|^{\mu}\leq CR^{2-(p+1)\mu/(p-1)}.\label{e2.4}
\end{gather}
\end{lemma}

\begin{proof}
  We can assume that $B_{R}$ is centered at $x_0\in \Omega$ and 
 first focus on proving \eqref{e2.3}. Let $\xi$ be  a $C^2$-cut-off
function  on $B_{2}$ satisfying:
\begin{enumerate}
\item $\xi(x)=\xi(|x-x_0|)$, $0\leq|x-x_0|\leq 2$.

\item $\xi(x)=1$ for $|x-x_0|\leq 1$.

\item $\xi$ has compact support in $B_{2}$ and $0 \leq \xi \leq 1 $.

\item $|\nabla \xi|\leq 2 $.
\end{enumerate}
Let $d=p-\gamma>0$ and $\phi=[\xi(\frac{x-x_0}{R})]^{k}u^{-d}$ as
a test function for \eqref{e2.2} ($k$ to be fixed later). We obtain
\[
-\int_{\Omega}(u_{rr}+u_{x_nx_n})\xi^{k}u^{-d}\geq
\int_{\Omega}(u^{p}-M|\nabla
u|^{\tau}+\frac{n-2}{r}u_{r})\xi^{k}u^{-d}.
\]
Integrating by parts and using that $|\nabla\xi^{k}|=k\xi^{k-1}|\nabla
\xi|\leq \xi^{k}\frac{2k}{R\xi}$, we obtain
\begin{align*}
&d\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+\int_{\Omega}\xi^{k}u^{\gamma}\\
&\leq\int_{\Omega}u^{-d}|\nabla
u||\nabla\xi^{k}|+M\int_{\Omega}|\nabla
u|^{\tau}\xi^{k}u^{-d}-\int_{\Omega}\frac{n-2}{r}u_{r}\xi^{k}u^{-d}
\\
&\leq\int_{\Omega}u^{-d}|\nabla
u|\xi^{k}\frac{2k}{R\xi}+M\int_{\Omega}|\nabla
u|^{\tau}\xi^{k}u^{-d}+\frac{n-2}{\operatorname{dist}(0,\partial\Omega)}\int_{\Omega}|\nabla
u|\xi^{k}u^{-d}.
\end{align*}
Applying the Young inequality to the first right term, we have
\[
\int_{\Omega}u^{-d}|\nabla
u|\xi^{k}\frac{2k}{R\xi}\leq\frac{d}{4}\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+CR^{-2}\int_{\Omega}\xi^{k-2}u^{\gamma-p+1},
\]
so
\begin{align*}
&\frac{3}{4}d\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+\int_{\Omega}\xi^{k}u^{\gamma}\\
&\leq CR^{-2}\int_{\Omega}\xi^{k-2}u^{\gamma-p+1}+M\int_{\Omega}|\nabla
u|^{\tau}\xi^{k}u^{-d}+\frac{n-2}{\operatorname{dist}(0,\partial\Omega)}\int_{\Omega}|\nabla
u|\xi^{k}u^{-d}.
\end{align*}
Next we focus on the  case of $\gamma>p-1$. Take
$k=\frac{2\gamma}{p-1}$. By using the Young inequality again, we
have
\[
CR^{-2}\int_{\Omega}\xi^{k-2}u^{\gamma-p+1}\leq
\frac{1}{4}\int_{\Omega}\xi^{k}u^{\gamma}+CR^{2-2\gamma/(p-1)}
\]
and
\begin{align*}
M\int_{\Omega}|\nabla
u|^{\tau}\xi^{k}u^{-d}
&\leq\frac{d}{4}\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+C\int_{\Omega}\xi^{k}u^{t}\\
&\leq\frac{d}{4}\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+\frac{1}{4}\int_{\Omega}\xi^{k}u^{\gamma}+CR^{-2},
\end{align*}
the second inequality holds becasue
$t=(-d-\tau\frac{\gamma-p-1}{2})\frac{2}{2-\tau}<\gamma$, and
\begin{align*}
\frac{n-2}{\operatorname{dist}(0,\partial\Omega)}\int_{\Omega}|\nabla
u|\xi^{k}u^{-d}
&\leq\frac{d}{4}\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+C\int_{\Omega}\xi^{k}u^{\gamma-p+1}\\
&\leq \frac{d}{4}\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+\frac{1}{4}\int_{\Omega}\xi^{k}u^{\gamma}+CR^{-2}.
\end{align*}
So
\begin{equation}
\frac{d}{4}\int_{\Omega}\xi^{k}u^{\gamma-p-1}|\nabla
u|^2+\frac{1}{4}\int_{\Omega}\xi^{k}u^{\gamma}\leq
CR^{2-2\gamma/(p-1)},\label{e2.5}
\end{equation}
which gives \eqref{e2.3}.

If  $\gamma=p-1$, \eqref{e2.3} is obvious by the above arguments. For the
case of $\gamma<p-1$, the following H\"{o}der inequality
\[
\int_{B_{R}}u^{\gamma}\leq
CR^{2(1-\gamma)/(p-1)}\Big(\int_{B_{R}}u^{p-1}\Big)^{\gamma/(p-1)}
\]
and the above argument yields to \eqref{e2.3}.

To prove \eqref{e2.4}, we use  H\"{o}der inequality:
\[
\int_{B_{R}}|\nabla u|^{\mu}\leq
\Big(\int_{B_{R}}u^{\gamma-p-1}|\nabla
u|^2\Big)^{\mu/2}\Big(\int_{B_{R}}u^{s}\Big)^{1-\frac{\mu}{2}},
\]
where $s=(p+1-\gamma)/(2-\mu)$. We  can choose $\gamma$ close enough
to $p-1$ such that $s<p$, and then obtain \eqref{e2.4} by combining \eqref{e2.3}
and \eqref{e2.5}. Thus we complete the proof.
\end{proof}

\begin{lemma} \label{lem2.2}
Let $u(r,x_n)$ be a nonnegative weak solution of the following
inequality, in a domain $\Omega$,
\[
|u_{rr}+u_{x_nx_n}|\leq c(x)|\nabla u| +d(x)u+f(x),
\]
where $c(x)\in L^{q'}(\Omega)$,
$d, f\in L^{q}(\Omega)$, $q'>2$ and $q\in (1,2)$.
Then for every  $R$ such that $B_{2R}\subset\Omega$,
there exists a constant
$C=C(q, q', R^{1-\frac{2}{q'}}\|c\|_{L^{q'}}$,
$R^{2-\frac{2}{q}}\|d\|_{L^{q}} )$ such that
\[
\sup_{B_{R}}    u\leq C(\inf_{B_{R}} u+R^{2-\frac{2}{q}}\|f\|_{L^{q}}).
\]
\end{lemma}

Note that this lemma is of Harnack type; see \cite{Serrin} for more
information on this type of inequalities.
The next theorem is similar to \cite[Theorem 2.3]{Ruiz}.

\begin{theorem} \label{thm2.2}
Let {\rm (D)}  hold and  $R\leq R_0$ such that $B_{2R}\subset\Omega$.
Suppose $u(r,x_n)$ is a positive weak solution of the inequality
\[
u^{p}-M|\nabla u|^{\tau}+\frac{n-2}{r}u_{r}\leq
-(u_{rr}+u_{x_nx_n})\leq c_0u^{p}+M|\nabla
u|^{\tau}+\frac{n-2}{r}u_{r}+\lambda,\label{e2.6}
\]
where $p>1$, $0<\tau<\frac{2p}{p+1}$, $\lambda>0$.  Then there
exists a constant $C=C(p, \tau, R_0, M)$ such that
\[
\sup_{B_{R}}  u\leq C(\inf_{B_{R}}   u+\lambda R^2).
\]
\end{theorem}

\begin{proof} From \eqref{e2.6}, we obtain
\[
|u_{rr}+u_{x_nx_n}|\leq c_0u^{p}+M|\nabla
u|^{\tau}+\frac{n-2}{r}|\nabla u|+\lambda.
\]
Take $f=\lambda$, $c=M|\nabla u|^{\tau-1}+\frac{n-2}{r}$ and
$d=c_0u^{p-1}$. To prove this theorem, we only need to verify that
\[
c(x)\in L^{q'}(B_{2R}), \quad d\in L^{q}(B_{2R}).
\]
Note that $\frac{n-2}{r}$ obviously belongs to $ L^{q'}(B_{2R})$,
so we only need to prove $M|\nabla u|^{\tau-1}\in L^{q'}(B_{2R})$.
By lemma 2.1, we have
\[
\|M|\nabla u|^{\tau-1}\|_{L^{q'}}=M\Big(\int_{B_{(2R)}}|\nabla
u|^{\mu}\Big)^{1/q'}\leq CR^{\frac{2-(p+1)\mu/(p-1)}{q'}},
\]
where $\mu=q'(\tau-1)$ should satisfy $q'(\tau-1)<\frac{2p}{p+1}$
for some $q'>2$. Since $\tau<\frac{2p}{p+1}$ and $q'>2$ can be close
enough to $2$, so we just need to verify
\[
2\big(\frac{2p}{p+1}-1\big)<\frac{2p}{p+1}.
\]
The above inequality is obvious, that is to say,
$c(x)\in L^{q'}(B_{2R})$.

For $d=c_0u^{p-1}$,  by lemma 2.1 we have
\[
\|d\|_{L^{q}(B_{2R})}=c_0\Big(\int_{B_{(2R)}}u^{\gamma}\Big)^{1/q}\leq
CR^{(2-2q)/q},
\]
where $\gamma=(p-1)q$ should satisfy $(p-1)q<p$. By choosing $q>1$
close enough to $1$, we can get  $(p-1)q<p$, that is, $d\in
L^{q}(B_{2R})$. The proof is complete.
\end{proof}

For  completeness, we sketch the proof of Theorem \ref{thm2.1} which is
similar as the proof of \cite[Proposition 3.3]{Ruiz}.

\begin{proof}[Proof of Theorem \ref{thm2.1}]
 Suppose, by contradiction,
that there exist $\lambda_n<\lambda_0$, $u_n>0$ such that
$u_n$ is solution of \eqref{e2.1} with $\lambda$ substituted by
$\lambda_n$ and $\max_{\Omega} u_n\to \infty$.
Let $z_n$ be a point in $\Omega$ such that
$u_n(z_n)=\max_{\Omega} u_n\triangleq S_n$. Denote
$\delta_n=\operatorname{dist}(z_n,\partial\Omega)$. In order to prove there
exists a $y_0\in \Omega$ such that
$u_n(y_0)\to\infty$, we proceed in three steps:
\smallskip

\noindent\textbf{Step 1:} There exists $c>0$ such that
$c<\delta_nS_n^{(p-1)/2}$. Define
$w(x)=S_n^{-1}u_n(y)$, where $y=M_nx+z_n$,
$M_n=S_n^{(1-p)/2}$. By easy computation and condition
(F2),  we obtain
\begin{align*}
-\Delta
w_n(x)
&=S_n^{-1}M_n^2(H(M_nx+z_n,S_nw_n(x),S_nM_n^{-1}\nabla
w_n(x))+\lambda_n)\\&\leq
c_0w_n^{p}+MS_n^{-p}S_n^{\tau\frac{p+1}{2}}|\nabla
w_n|^{\tau} +\frac{n-2}{\operatorname{dist}(0,\partial\Omega)}|\nabla
w_n|+\lambda_nS_n^{-p}.
\end{align*}
Notice that $MS_n^{-p}S_n^{\tau\frac{p+1}{2}}$  and
$\lambda_nS_n^{-p}$ tend to zero respectively  as $n$ tends to
infinity, so
\[
-\Delta w_n(x)\leq c_0w_n^{p}+|\nabla
w_n|^{\tau}+\frac{n-2}{\operatorname{dist}(0,\partial\Omega)}|\nabla w_n|+1.
\]
By the regularity  result in \cite{Lieberman}, there exists a
constant $C$ independent of $n$ such that
$\sup_{\Omega} w_n\leq C$. Let $y_n\in \partial\Omega$ such that
$d(z_n,y_n)=\delta_n$; then, by the mean value theorem, we have
\[
1=w_n(0)=w_n(0)-w_n(M_n^{-1}(y_n-z_n))\leq
\sup_{\Omega} w_nM_n^{-1}\delta_n\leq
CM_n^{-1}\delta_n.
\]
Thus, the first step is complete.
\smallskip

\noindent\textbf{Step 2:} There exists $\gamma>0$ such that
\[
\int_{B(z_n,\delta_n/2)}|u_n|^{\gamma}\to\infty.
\]
By Theorem \ref{thm2.2}, we obtain
\[
S_n=\max_{B(z_n,\delta_n/2)} u_n\leq
C\Big(\min_{B(z_n,\delta_n/2)}
u_n+\lambda_n\frac{\delta_n^2}{4}\Big).
\]
Since $\lambda_n$ and $\delta_n$ are bounded, we obtain that
$\min_{B(z_n,\delta_n/2)} u_n\geq cS_n$ for
some $c>0$. So
\[
\int_{B(z_n,\delta_n/2)}|u_n|^{\gamma}\geq
cS_n^{\gamma}\delta_n^2\geq cS_n^{\gamma}S_n^{1-p}.
\]
We can choose a $\gamma>p-1$ such that
$cS_n^{\gamma}S_n^{1-p}\to+\infty$. The proof of step 2
is complete.
\smallskip

\noindent\textbf{Step 3:} There exists a $y_0\in \Omega$ such that
$u_n(y_0)\to\infty$. Notice that $\partial\Omega$ is
$C^2$ and compact boundary , so we can find $\varepsilon>0$
independent of $n$ and $y_n\in\Omega$ such that:
\begin{itemize}
\item $d(y_n,\partial\Omega)=2\varepsilon$, for all $n\in \mathbb{N}$.

\item $B(z_n,\frac{\delta_n}{2})\subset B(y_n,2\varepsilon)$, for all 
$n\in \mathbb{N}$.
\end{itemize}
By the weak Harnack inequality  in \cite{Trudinger} and step 2, we
conclude that
\[
\min_{B(y_n,\varepsilon)} u_n\geq
c\Big(\int_{B(y_n,2\varepsilon)}|u_n|^{\gamma}\Big)^{1/\gamma}\to+\infty.
\]
Taking a subsequence $\{\tilde{y}_n\}\subset \{y_n\}$ such that
$\tilde{y}_n\to y_0\in \Omega$. For $n$ large enough, we
have $y_0\in B(\tilde{y}_n,\varepsilon)$ and
$u_n(y_0)\to\infty$, which contradicts with Theorem \ref{thm2.2}.
Thus we obtain a priori estimate of solutions.
\end{proof}

\section{Existence of positive classical $C^{2,\beta}$-solutions}

\begin{theorem} \label{thm3.1}
Assume {\rm (D1), (F1)--(F3)} hold. Then \eqref{e1.1} admits an $(n-1)$-radial
symmetric positive classical solution
 $u(r,x_n)\in C^{2,\beta}(\Omega)\cap C^{0}(\overline{\Omega})$.
\end{theorem}

The following lemma mentioned in \cite{Gilbarg} will be used in our
proof.


\begin{lemma}[{\cite[Theorem 12.4]{Gilbarg}}]  \label{lem3.1}
Let $u$ be a bounded $C^2(\Omega)$ solution of
\[
Lu=a(x,y)u_{xx}+2b(x,y)u_{xy}+c(x,y)u_{yy}=f(x,y),
\]
where $L$ is uniformly elliptic in a domain
$\Omega\subset\mathbb{R}^2$, satisfying
\begin{gather*}
\lambda(\xi^2+\eta^2)\leq
a\xi^2+2b\xi\eta+c\eta^2\leq\Lambda(\xi^2+\eta^2), \quad
\forall (\xi,\eta)\in \mathbb{R}^2, \\
\frac{\Lambda}{\lambda}\leq \gamma
\end{gather*}
for some constant $\gamma\geq 1$. Then for some
$\alpha=\alpha(\gamma)>0$, we have
\[
[u]_{1,\alpha}^{*}=\sup_{z_{1},z_{2}\in \Omega}
d_{1,2}^{1+\alpha}\frac{|Du(z_{2})-Du(z_{1})|}{|z_{2}-z_{1}|^{\alpha}}\leq
C(|u|_0+|\frac{f}{\lambda}|_0^{(2)}),
\]
where $C=C(\gamma)$, $|\frac{f}{\lambda}|_0^{(2)}
=\sup_{z\in \Omega} d_{z}^2|\frac{f}{\lambda}|$,
$d_{z}=\operatorname{dist}(z,\partial\Omega)$ and
$d_{1,2}=min\{d_{z_{1}},d_{z_{2}}\}$.
\end{lemma}

Since the conditions imposed on $f$ in Theorem \ref{thm3.1} are different
from those in \cite[Theorem 12.5]{Gilbarg}, 
it is necessary to  give the proof,  although similar to that of
\cite[Theorem 12.5]{Gilbarg}.

\begin{proof}[Proof of Theorem \ref{thm3.1}]
We now proceed by truncation of $H$ to reduce \eqref{e1.4} to the case 
of bounded $H$. Namely, let $\psi_{N}$ denote the function given by
\[
\psi_{N}(t)=\begin{cases}
t, & |t|\leq N\\
N \operatorname{sign} t, &|t|>N,
\end{cases}
\]
and define the truncation of $H$ by
\[
H_{N}(r,x_n,u,u_{r},u_{x_n})=H(r,x_n,\psi_{N}(u),\psi_{N}(u_{r}),
\psi_{N}(u_{x_n})).
\]
 From (F2), we have $|H_{N}|\leq c_0N^{p}+MN^{\tau}+\frac{n-2}{\operatorname{dist}
(0,\partial\Omega)}N=C_0$.  Consider
now the family of problems
\begin{equation}
\begin{gathered}
-(u_{rr}+u_{x_nx_n})=H_{N}(r,x_n,u,u_{r},u_{x_n})\quad  \text{in } \Omega ,\\
\ \ \ u(x)=0 , \quad \text{on } \partial\Omega.
\end{gathered}\label{e3.1}
\end{equation}
By Theorem \ref{thm2.1}, any solution u of \eqref{e3.1} is subject to the
bound $\tilde{M}$, independent of $N$,
\begin{equation}
\sup_{\Omega} |u|\leq \tilde{M}.\label{e3.2}
\end{equation}
Now we make the following observation. Let $v$ be any bounded
function with locally H\"older continuous first derivatives
in $\Omega$ and $\tilde{H_{N}}=H_{N}(r,x_n,v,v_{r},v_{x_n})$.
Then the following linear problem
\begin{equation}
\begin{gathered}
-(u_{rr}+u_{x_nx_n})=\tilde{H_{N}}\quad  \text{in } \Omega ,\\
u(x)=0 , \quad \text{on } \partial\Omega,
\end{gathered}\label{e3.3}
\end{equation}
has a unique solution $u\in C^2(\Omega)\cap C^{0}(\bar{\Omega})$.
We observe from classical priori estimates  that
\[
|u|_0=\sup_{\Omega} |u|\leq M_0.
\]
 Furthermore,
if $\sup_{\Omega}|v|\leq M_0 $, from lemma 3.1, we have
\[
|u|_{1,\alpha}^{*}\leq C(|u|_0+C_0(\operatorname{diam}(\Omega))^2)\leq
C(M_0+C_0(\operatorname{diam}(\Omega))^2)=K,
\]
where $C$, $\alpha$  depend on $M_0$. So $K$ depends on $M_0, N$
and $\Omega$.

Next, define  the Banach space
\[
C^{1,\alpha}_{*}(\Omega)=\{u\in
C^{1,\alpha}(\Omega)||u|_{1,\alpha;\Omega}^{*}<+\infty\}
\]
and define a mapping $T$ on the set
\[
\mathbb{S}=\{v\in C^{1,\alpha}_{*}:|v|_{1,\alpha}^{*}\leq K,|v|_0\leq M_0\}.
\]
So $u= Tv$ is the unique solution of the linear Dirichlet problem
\eqref{e3.3}. It is easy to show that $\mathbb{S}$ is convex and  closed in
the Banach space, and $T$ is continuous in
$C_{*}^1=\{u\in C^1(\Omega)||u|_{1;\Omega}^{*}<+\infty$ and
$T\mathbb{S}$ is precompact. So we may conclude from the Schauder
fixed point theorem and Schauder estimates that T has a fixed point,
$u_{N}= Tu_{N}$, $u_{N}\in C^{1,\alpha}_{*}(\Omega)\cap C^{2,\beta}(\Omega)\cap
C^{o}(\bar{\Omega})$. This will provide a solution of the problem
\eqref{e3.1}.

Furthermore, from lemma 3.1 we infer the estimate
\[
[u_{N}]^{*}_{1,\alpha}\leq C(|u|_0+|G_{HN}|_0^{(2)}).
\]
By (F2) and \eqref{e3.2}, we obtain
\[
[u_{N}]^{*}_{1,\alpha}\leq C(1+[u_{N}]_{1}^{*}),
\]
where $C=C(\tilde{M},M,c_0,p,\tau, \operatorname{diam}(\Omega))$.
 Furthermore, the interpolation inequality yields the uniform bound which is
independent of $N$,
\[
[u_{N}]^{*}_{1,\alpha}\leq C=C(\tilde{M},M,c_0,p,\tau,
 \operatorname{diam}(\Omega)).
\]

By similar arguments as in  the proof of \cite[Theorem 12.5]{Gilbarg}, 
it is easy to show there is a subsequence
$\{u_n\}$ of $\{u_{N}\}$ which converges to a solution u of \eqref{e1.4},
and $u$ also satisfies the boundary condition $u=0$ on
$\partial\Omega$. Since $f$ is nonnegative, by comparison
principles, $u$ is positive. This completes the proof.
\end{proof}


\begin{remark} \label{rmk1} \rm
If $\Omega=\Omega_{1}\times\Omega_{2}\subset\mathbb{R}^{k}\times\mathbb{R}^{n-k}$,
 $\Omega_{1}$ and $\Omega_{2}$ are symmetric
and $0\notin\overline{\Omega}$, 
$f(x,u,|\nabla u|)=f(r_{1},r_{2},u,|\nabla u|)$, where
$r_{1}=\sqrt{x_{1}^2+x_{2}^2+\cdots+x_k^2}$,
$r_{2}=\sqrt{x_{k+1}^2+x_{k+2}^2+\cdots+x_n^2}$.  Under the
conditions of (F2) and (F3),  \eqref{e1.1} admits an  $(n-1)$-radial
symmetric positive classical solution
$u(r_{1},r_{2})\in C^{2,\beta}(\Omega)\cap C^{0}(\overline{\Omega})$. 
The proof is left to readers. 
\end{remark}

\subsection*{Acknowledgments}
This work is partly supported by the National Natural Science
Foundation of China (10971088) and Natural Science Foundation of
Chizhou College (2013ZRZ002).


\begin{thebibliography}{00}

\bibitem{Abdellaoui} B. Abdellaoui,  A. Dall Aglio,  I. Peral; 
\emph{Some remarks on elliptic problems with
critical growth in the gradient}, J. Differential Equations 222
(2006), 21--62.

\bibitem{Alves} Claudianor O. Alves, Paulo C. Carriao, Luiz F. O. Faria;
\emph{Existence of solutions to singular elliptic equations with convection 
terms via the Galerkin method}, Electronic Journal of Differential Equations
Vol. 2010 (2010), No. 12, 1--12.


\bibitem{Amann} H. Amann, M. G. Crandall;
\emph{On some existence theorems for semilinear
elliptic equations}, Indiana Univ. Math. J 27 (1978), 779--790.

\bibitem{Bisci} Giovanni Molica Bisci, Vicentiu R\u{a}dulescu;
\emph{Multiple symmetric solutions for a Neumann problem with lack of
compactness}, C. R. Acad. Sci. Paris, Ser. I 351 (2013) 37--42.

\bibitem{Chen Wenjing} Wenjing Chen, Jianfu Yang;
\emph{Existence of positive solutions for quasilinear elliptic equation 
on Riemannian manifolds}, 
Differential Equations and Applications Vol 2 (2010), 569--574.

\bibitem{Figueiredo1} D. G. de Figueiredo, M. Girardi, M. Matzeu;
\emph{Semilinear ellptic equations with dependence on the gradient via
mountain-pass techniques},  Differential and Integral Equations 17
(2004), 119--126.

\bibitem{Figueiredo2} D. G. de Figueiredo, J. S\'anchez, P. Ubilla;
\emph{Quasilinear equations with dependence on the gradient}, Nonlinear
Analysis 71 (2009), 4862--4868.

\bibitem{Ghergu1} M. Ghergu, V. R\u{a}dulescu;
\emph{Bifurcation for a class of singular elliptic problems with quadratic 
convection term}, C. R. Acad. Sci. Paris, Ser. I 338 (2004),  831--836.

\bibitem{Ghergu2} M. Ghergu, V.R\u{a}dulescu; 
\emph{On a class of sublinear singular elliptic problems with convection term}, 
J. Math. Anal. Appl. 311 (2005) 635--646.

\bibitem{Gilbarg} D.Gilbarg, N. S. Trudinger;
\emph{Elliptic Partial Differential Equations of Second Oder}, 
second ed. Springer-Verlag, Berlin, 1983.

\bibitem{Girardi} M. Girardi, M. Matzeu;
\emph{Positive and negative solutions of a quasilinear elliptic equation 
by a Mountain Pass method and truncature techniques}, 
Nonlinear Analysis T.M.A. 59 (2004), 199--210.

\bibitem{Lieberman} G. M. Lieberman;
\emph{Boundary regularity for solutions of degenerate elliptic equations}, 
Nonlinear Anal. 12 (1988) 1203--1219.

\bibitem{Pohozaev} Pohozaev S;
\emph{On equations of the type $\Delta u=f(x,u,Du)$}, Mat. Sb. 113 (1980),
324--338.

\bibitem{Ruiz} D. Ruiz; 
\emph{A priori estimates and existence of positive solutions for strongly
 nonlinear problems},  J. Differential Equations 199 (2004), 96--114.

\bibitem{Serrin} J. Serrin, H. Zou;
\emph{Cauchy-Liouville and universal boundedness theorems for quasilinear 
elliptic equations and inequalities}, Acta Math. 189 (2002) 79--142.

\bibitem{Trudinger} N. Trudinger;
\emph{On Harnack type inequalities and their applications to quasilinear 
elliptic equations}, Comm. Pure Appl. Math. 20 (1967) 721--747.

\bibitem{Wang} X. Wang, Y. Deng;
\emph{Existence of multiple solutions to nonlinear
elliptic equations in nondivergence form}, 
J. Math. Anal. and Appl. 189 (1995), 617--630.

\bibitem{Xavier} J. B. M. Xavier;
\emph{Some existence theorems for equations of the form $-\Delta
u=f(x,u,Du)$}, Nonlinear Analysis T.M.A. 15 (1990), 59--67.

\bibitem{Yan} Z. Yan;
\emph{A note on the solvability in $W^{2,p}(\Omega)$ for the equation
$-\Delta u=f(x,u,Du)$}, Nonlinear Analysis T.M.A. 24 (1995),
1413--1416.

\bibitem{Zou} Henghui Zou;
\emph{A priori estimates and existence for
quasilinear elliptic equations} 
Calc. Var. Partial Differential Equations 33 (2008), no. 4, 417--437. 

\end{thebibliography}

\end{document}