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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 53, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/53\hfil Ground state solutions]
{A remark on ground state solutions for Lane-Emden-Fowler equations
 with \\ a convection term}

\author[H. Xue, Z. Zhang\hfil EJDE-2007/53\hfilneg]
{Hongtao Xue, Zhijun Zhang}  % in alphabetical order

\address{Hongtao Xue \newline
School of Mathematics and Informational Science,
Yantai University, Yantai, Shandong, 264005, China}
\email{ytxiaoxue@yahoo.com.cn}

\address{Zhijun Zhang \newline
 School of Mathematics and Informational Science,
Yantai University, Yantai, Shandong, 264005, China}
\email{zhangzj@ytu.edu.cn}

\thanks{Submitted January 6, 2007. Published April 10, 2007.}
\thanks{Z. Zhang is supported by grant 10671169 from NNSFC}
\subjclass[2000]{35J60, 35B25, 35B50, 35R05}
\keywords{Ground state solution; Lane-Emden-Fowler equation; \hfill\break\indent
convection term; maximum principle; existence; sub-solution; super-solution}

\begin{abstract}
 Via  a sub-supersolution  method and  a perturbation argument, we
 study the Lane-Emden-Fowler equation
 $$
 -\Delta u =p(x)[g(u)+f(u)+|\nabla u|^q]
 $$
 in $\mathbb{R} ^N$ ($N\geq3$), where $0<q<1$, $p$ is a positive
 weight such that $\int_0^\infty r\varphi(r)dr<\infty$, where
 $\varphi(r)=\max_{|x|=r}p(x)$, $r\geq 0$. Under the
 hypotheses that both $g$ and $f$ are sublinear, which  include no
 monotonicity on the functions $g(u)$,  $f(u)$, $g(u)/u$ and
 $f(u)/u$, we show the existence of ground state solutions.
\end{abstract}

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

\section{Introduction}

This paper concerns  the Lane-Emden-Fowler type problem
\begin{equation}
\begin{gathered}
-\Delta u =p(x)[g(u)+f(u)+|\nabla u|^q],  \quad\text{in }\mathbb{R}^N,\\
u>0,  \quad\text{in }\mathbb{R}^N,\\
u(x)\to 0,  \quad \text{ as } |x|\to \infty,
\end{gathered}
 \label{e1.1}
\end{equation}
where $N\geq3$, $0<q<1$, and $p:\mathbb{R}^N\to (0,+\infty)$
is a locally H\"{o}lder continuous function of exponent
$0<\alpha<1$ satisfying
\begin{equation}
\int_0^\infty r\varphi(r)dr<\infty,\label{e1.2}
\end{equation}
where $\varphi(r)=\max_{|x|=r}p(x)$, $r\geq 0$. We also assume
that $g$ satisfies
\begin{itemize}
\item [(G1)] $g\in C^1((0,\infty),(0,\infty))$;
\item [(G2)] $\lim_{u\to  0^+} \frac{g(u)}{u}=+\infty$;
\item [(G3)] $\lim_{u\to  \infty}\frac{g(u)}{u}=0$,
\end{itemize}
and $f:[0,\infty)\to [0,\infty)$ is a locally H\"{o}lder
continuous function of exponent $0<\alpha<1$   satisfying
\begin{itemize}
\item [(F1)] $\lim_{u\to  0^+} \frac{f(u)}{u}=+\infty$;
\item [(F2)] $\lim_{u\to \infty} \frac{f(u)}{u}=0$.
\end{itemize}


The Lane-Emden-Fowler equation arises in the study of gaseous
dynamics in astrophysics, fluid mechanics, relativistic mechanics,
nuclear physics and  chemical reaction systems. By far, it has
been  studied by many authors using various methods.  But we note
that in most works, monotonicity is necessary to some extent.

With regard to semilinear  elliptic problems in bounded domains,
we refer for details to \cite{c2, c3, g5, s1, s2,  z4} and their
references. Here, we mention the works of Ghergu and
R\v{a}dulescu \cite{ g1}, Zhang \cite{z1, z3}, where the influence
of the convection term has been emphasized.



Concerning with ground state solutions for  elliptic problems,
that is, positive solutions defined in the whole space and
decaying to zero at infinity, we refer the reader to the works of
C\v{i}rstea and R\v{a}dulescu \cite{c1}, Dinu \cite{d1, d2},
Goncalves and Santos \cite{g4}, Sun and Li \cite{s3}, Ye and Zhou
\cite{y1}, Zhang \cite{z2}. We mention here the work of Zhang
\cite{z5}. In \cite{z5}, it showed  that if $g$ satisfies
(G1)-(G3) and condition \eqref{e1.2}, then the following boundary value
problem
\begin{equation}
\begin{gathered}
-\Delta u =p(x)g(u),  \quad\text{in }\mathbb{R}^N,\\
u>0,  \quad\text{in }\mathbb{R}^N,\\
u(x)\to 0,  \quad\text{as }  |x|\to \infty,
\end{gathered} \label{e1.3}
\end{equation}
has at least one solution $ u\in C^{2+\alpha} _ {\rm loc} (\mathbb{R}^N)$.
For problem \eqref{e1.3},  we see that no monotonicity conditions
are imposed on the functions  $g(u)$ and   $g(u)/u$ . On the other
hand, condition \eqref{e1.2} is necessary to prove the existence (see
also Lair and Shaker \cite{l2}).


Recently, in Ghergu and R\v{a}dulescu \cite{g2},  the same problem
\eqref{e1.1} is considered, where $g\in C^1(0,\infty)$ is a positive
decreasing function such that
 $$
\lim_{u\to  0^+}g(u)=+\infty,
$$
  and
$f:[0,\infty)\to [0,\infty)$ is a H\"{o}lder continuous
function of exponent $0<\alpha<1$ which is non-decreasing such
that  $f>0$ on $(0,\infty)$  and satisfies (F1)-(F2) and
\begin{itemize}
\item [(F3)] the mapping $(0,\infty)\ni u\longmapsto\frac{f(u)}{u}$
is non-increasing.
\end{itemize}
Finally, they showed that in addition to condition \eqref{e1.2} , if the
above assumptions are fulfilled, then problem \eqref{e1.1} has at least
one ground state solution.

In the present  paper, we  consider the existence of
ground state solutions for problem \eqref{e1.1} under  more general
conditions.
 Our main result is summarized in the following theorem.

\begin{theorem}\label{thm1.1}
 Assume  (G1)--(G3) and (F1)--(F2).
 Then problem \eqref{e1.1} has at least one  solution provided
 that condition \eqref{e1.2} is fulfilled.
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
 Some basic examples of the function $g$ satisfying (G1)--(G3) are:

\begin{itemize}
\item [(i)]   $u^{-\gamma}+u^p+ \sin {\psi(u)}+1$, where $\gamma >0$, $p< 1$
 and $\psi\in C^2(\mathbb{R})$;

\item [(ii)] $e^{1/{u^\gamma}}+u^p+\cos {\psi(u)}+1$, where $\gamma >0$,
$p< 1$ and  $\psi\in C^2(\mathbb{R})$;

 \item [(iii)] $u^{-\gamma}\ln^{-q_1}(1+u)+\ln^{q_2}(1+u)+u^p+\sin
{\psi(u)}+2$ with $\psi\in C^2(\mathbb{R})$ , $\gamma >0$, $p<1$,
$q_2>0$ and $q_1>0$;

\item [(iv)] $u^{-\gamma}+\arctan {\psi(u)}+\pi$ with $\psi\in C^2(\mathbb{R})$ and $\gamma >0$.
\end{itemize}
\end{remark}

\begin{remark} \label{rmk1.3} \rm
 Some basic examples of the function $f$
satisfying (F1)--(F2)  are:
\begin{itemize}
\item [(i)] $c_1(1+u)^{-\alpha}+c_2u^\gamma+c_3$,   where $c_1, c_2,
c_3\geq0$, $\alpha>0$, $0<\gamma<1$;

\item [(ii)] $e^{1/{(u+1)}}+u^\gamma+\sin{\psi(u)}+1$, where $0<\gamma<1$
and $\psi\in C^2(\mathbb{R})$;

\item [(iii)] $\ln^q(u+1)+(1+u)^{-\alpha} $, where $q>0$, $\alpha>0$;

\item [(iv)] $u^p\ln^q(u+1) $, where $p, q\in (0, 1)$ and $p+q<1$.
\end{itemize}
\end{remark}


\section{Proof of Theorem \ref{thm1.1}}

In this section, we first show the existence of positive solutions
for  problem \eqref{e1.1} in smooth bounded domains by a
sub-supersolution method. Then, via the perturbation argument, we
prove Theorem \ref{thm1.1}.
 First we recall the following auxiliary results.


\begin{lemma}[{\cite[Lemma 3]{c3}}] \label{lem2.1}
Let $\Omega\subset\mathbb{R}^N$ be a smooth bounded domain. Suppose the
boundary-value problem
\begin{equation}
-\Delta u =p(x)[g(u)+f(u)+|\nabla u|^q], \quad u>0,\; x\in\Omega,
\; u|_{\partial\Omega}=0,  \label{e2.1}
\end{equation}
has a super-solution $\bar{u}$ and a
 sub-solution $\underline{u}$ such that $\underline{u}\leq\bar{u}$ in
$\Omega$, then  problem \eqref{e2.1}  has at least one solution
$u\in C(\bar{\Omega})\cap C^2(\Omega)$ in the ordered
 interval $[\underline{u}, \bar{u}]$.
\end{lemma}



\begin{lemma}[{\cite [Lemma 2.3]{z5}}] \label{lem2.2}
 Suppose (G1)--(G3) are satisfied. Then there exists a function
 $\overline{g}_0$ such that
\begin{itemize}
\item [(i)]  $\overline{g}_0 \in C^1((0,\infty),(0,\infty))$;
\item [(ii)] $\frac{g(s)}{s}\leq\overline{g}_0, \forall\ s>0$;
\item [(iii)]  $\overline{g}_0(s)$  is non-increasing  on
$(0,\infty)$;
\item [(iv)]$\lim_{s\to 0+}\overline{g}_0(s)=\infty$ and
$\lim_{s\to \infty}\overline{g}_0(s)=0$.
\end{itemize}
\end{lemma}

 Note that if $g\in C((0,\infty),(0,\infty))$,
   the  function $\overline{g}_0$ still exists in Lemma \ref{lem2.2}.

\begin{lemma} \label{lem2.3}
 Let $\Omega\subset\mathbb{R}^N$ be a smooth
bounded domain. Assume (G1)-(G3) and (F1)-(F2) are fulfilled. Then
problem \eqref{e2.1} has at least one solution
$u\in C(\bar{\Omega})\cap C^2(\Omega)$.
\end{lemma}

\begin{proof}
 Let $\underline{u}$ be a solution of
\begin{equation}
-\Delta \underline{u} =p(x)g(\underline{u}), \quad \underline{u}>0,
\; x\in\Omega, \quad \underline{u}|_{\partial\Omega}=0. \label{e2.2}
\end{equation}
The existence of $\underline{u}$ follows from the results in Zhang
\cite{z5}. Obviously, $\underline{u}$ is a sub-solution of \eqref{e2.1}.
The main point is to find a super-solution $\overline{u}$ of \eqref{e2.1}
such that $\underline{u}\leq\overline{u}$ in $\Omega$. Then, by
Lemma \ref{lem2.1} we deduce that problem \eqref{e2.1} has at least one
solution.

Denote $\sigma(u):=g(u)+f(u)$.  Then $\sigma$ satisfies
\begin{itemize}
\item %[($\sigma1$)]
$\sigma\in C((0,\infty),(0,\infty))$;
\item %[($\sigma2$)]
$\lim_{u\to  0^+}\frac{\sigma(u)}{u}=+\infty$;
\item %[($\sigma3$)]
$\lim_{u\to  \infty}\frac{\sigma(u)}{u}=0$.
\end{itemize}
By Lemma \ref{lem2.2}, corresponding to $\sigma$,  there exists a function
$\overline{\sigma}_0$ satisfying
\begin{itemize}
\item [(i)]  $\overline{\sigma}_0 \in C^1((0,\infty),(0,\infty))$;
\item [(ii)] $\frac{\sigma(u)}{u}\leq\overline{\sigma}_0$, for all
$u>0$;
\item [(iii)]  $\overline{\sigma}_0(u)$  is non-increasing
on $(0,\infty)$;
\item [(iv)] $\lim_{u\to 0+}\overline{\sigma}_0(u)=+\infty$ and
$\lim_{u\to \infty}\overline{\sigma}_0(u)=0$,
\end{itemize}
such that
 $G(u):=u(\overline{\sigma}_0(u)+\frac{1}{u})$ satisfying
 \begin{itemize}
\item[(G1)] $G\in C^1((0,\infty),(0,\infty))$; \item[(G2)]
$\frac{G(u)}{u}$ is decreasing on $(0,\infty)$;
\item[(G3)]$\lim_{u\to  0^+} \frac{G(u)}{u}=\infty$;
\item[(G4)]  $\lim_{u\to  \infty}\frac{G(u)}{u}=0$.
\end{itemize}
Then, we consider the  problem
\begin{equation}
-\Delta u =p(x)[G(u)+|\nabla u|^q],\quad u>0, \quad x\in\Omega, \quad
u|_{\partial\Omega}=0.
 \label{e2.3}
\end{equation}
We claim that this problem has at least one classical solution,
which is a super-solution of \eqref{e2.1}.
Indeed, let $h:[0,\eta]\to [0,\infty)$ be such that
\begin{equation}
\begin{gathered}
-h''(t) =\frac{G(h(t))}{h(t)}, \quad 0<t<\eta<1,\\
h>0, \quad 0<t\leq\eta<1,\\
h(0)=0.
\end{gathered} \label{e2.4}
\end{equation}
The existence of $h$ follows from the results in Agarwal and
O'Regan \cite[Theorem 2.1]{a1}. Since $h$ is concave, there exists
$h'(0+)\in(0,\infty]$, namely,  $h$ is increasing on $[0,\eta]$
for $\eta>0$ small enough. Multiplying by $h'(t)$ in \eqref{e2.4} and
integrating on $[t,\eta]$, we combine (G2) and get
$$
(h')^2(t)\leq 2h(\eta)\frac{G(h(t))}{h(t)}+(h')^2(\eta), \quad 0<t<\eta.
$$
Since $s^q\leq s^2+1$, for all $s\geq0$. Combining the above
inequality  we have
\begin{equation}
(h')^q(t)\leq C\frac{G(h(t))}{h(t)}, \label{e2.5}
\end{equation}
for all $ 0<t<\eta<1$ and some $C>0$.


Let $\phi_1$ be the normalized positive eigenfunction
corresponding to the first eigenvalue $\lambda_1$
 of $-\Delta$ in $H_0^1(\Omega)$.
By H\"{o}pf's maximum principle, there exist $\delta>0$ and
$\omega\Subset\Omega$ such that
\begin{equation}
|\nabla\phi_1|>\delta,\quad \text{in }  \Omega\setminus\omega.\label{e2.6}
\end{equation}

For the rest of this paper we denote
\begin{gather*}
|\phi_1|_\infty:=\max_{x\in\bar{\Omega}}\phi_1(x),\quad
|\phi_1|_0:=\min_{x\in\bar{\omega}}\phi_1(x), \\
|p|_{\infty}:=\max_{x\in\bar{\Omega}}p(x),\quad
|\nabla\phi_1|_\infty:=\max_{x\in\bar{\Omega}}|\nabla\phi_1(x)|.
\end{gather*}
 And we fix $c>0$ such that $c|\phi_1|_\infty<\eta$.

Using the monotonicity of  $h$ and $h'$, it follows that
\begin{gather}
0<h(c|\phi_1|_0)\leq h(c\phi_1)\leq h(\eta),\quad \text{in }\omega,
\label{e2.7}\\
0<h'(\eta)\leq h'(c\phi_1) \leq  h'(c|\phi_1|_0), \quad \text{in }\omega.
\label{e2.8}
\end{gather}
Let $M>1$ be such that
\begin{gather}
\lambda_1(Mch'(\eta))^{1-q}|\phi|_0>2|p|_{\infty}|{\nabla\phi_1|^q_\infty},
\label{e2.9}\\
M^{1-q}C^{-1}(c\delta)^{2-q}>2|p|_{\infty}.\label{e2.10}
\end{gather}
By (G4), we can choose $M>1$ large enough such that
\begin{equation}
\frac{G(Mh(c|\phi_1|_0))}{Mh(c|\phi_1|_0)}
 \leq\frac{\lambda_1c|\phi_1|_0h'(\eta)}{2|p|_{\infty}h(\eta)}.\label{e2.11}
\end{equation}
Next, we show that $\overline{u}_0=Mh(c\phi_1)$ is a
super-solution of  \eqref{e2.3} provided that $M$ satisfies
\eqref{e2.9}-\eqref{e2.11}. We have
$$
-\Delta \overline{u}_0=\lambda_1Mc\phi_1 h'(c\phi_1)
+Mc^2|\nabla\phi_1|^2\frac{G(h(c\phi_1))}{h(c\phi_1)}.
$$\label{e2.12}
By (G2), \eqref{e2.7}-\eqref{e2.8} and \eqref{e2.11}, we have
$$
\frac{G(Mh(c\phi_1))}{Mh(c\phi_1)}
\leq\frac{G(Mh(c|\phi_1|_0))}{Mh(c|\phi_1|_0)}
\leq\frac{\lambda_1c|\phi_1|_0h'(\eta)}{2|p|_{\infty}h(\eta)}
\leq\frac{\lambda_1c\phi_1h'(c\phi_1)}{2p(x)h(c\phi_1)},\quad\text{in }\omega.
$$%\label{e2.13}
It follows that
\begin{equation}
\lambda_1Mc\phi_1 h'(c\phi_1)\geq2p(x)G(Mh(c\phi_1)),
 \quad\text{in } \omega.
\label{e2.14}
\end{equation}
From \eqref{e2.8}-\eqref{e2.9}, we have
\begin{equation}
\lambda_1Mc\phi_1 h'(c\phi_1)\geq2p(x)|Mch'(c\phi_1)\nabla\phi_1|^q
=2p(x)|\nabla\overline{u}_0|^q,\  {\rm in}   \quad \omega.\label{e2.15}
\end{equation}
Since $h(0)=0$, we get
$$
\lim_{x\to \partial\Omega}\Big(\frac{(c\delta)^2}{h(c\phi_1)}
-2|p|_{\infty}\Big)=+\infty,
$$
namely,
\begin{equation}
\frac{(c\delta)^2}{h(c\phi_1)}>2|p|_{\infty}>2p(x), \quad\text{in }
\Omega\setminus\omega.\label{e2.16}
\end{equation}
From \eqref{e2.6}, (G2) and \eqref{e2.16}, we have
\begin{equation}
Mc^2|\nabla\phi_1|^2\frac{G(h(c\phi_1))}{h(c\phi_1)}
\geq Mc^2\delta^2\frac{G(Mh(c\phi_1))}{Mh(c\phi_1)}
\geq2p(x)G(Mh(c\phi_1)), \quad\text{in }  \Omega\setminus\omega.
\label{e2.17}
\end{equation}
From \eqref{e2.5}-\eqref{e2.6} and \eqref{e2.10}, we have
\begin{equation}
Mc^2|\nabla\phi_1|^2\frac{G(h(c\phi_1))}{h(c\phi_1)}
\geq 2p(x)|Mch'(c\phi_1) \nabla\phi_1|^q=2p(x)|\nabla\bar{u}_0|^q,\quad
\text{in }\Omega\setminus\omega.\label{e2.18}
\end{equation}
 From \eqref{e2.14}-\eqref{e2.15} and \eqref{e2.17}-\eqref{e2.18},
 we deduce that
$$
-\Delta \overline{u}_0\geq p(x)[G(\overline{u}_0)+|\nabla\overline{u}_0|^q],
\quad\text{in } \Omega,
$$
namely, $\overline{u}_0=Mh(c\phi_1)$ is a super-solution of
 \eqref{e2.3}.

On the other hand, the unique solution $\underline{u}_0$ of the
 boundary-value problem
$$
-\Delta \underline{u}_0 =p(x)G(\underline{u}_0),\quad
\underline{u}_0>0,\; x\in\Omega,\;
\underline{u}_0|_{\partial\Omega}=0,
% \label{e2.19}
$$
is a sub-solution of problem \eqref{e2.3}. Here the existence of
$\underline{u}_0$ follows from the results in  Goncalves and
Santos \cite{g4}.


Next, we prove that
$$
\underline{u}_0\leq\bar{u}_0 \quad\text{in } \Omega.
$$
Assume the contrary; i.e.,  there exists
 $x_0\in \Omega$ such that $\bar{u}_0(x_0)<\underline{u}_0(x_0)$. Then,
$\sup_{x\in \Omega} \left (\ln(\underline{u}_0(x))-\ln
(\bar{u}_0(x))\right )$ exists and is positive in $\Omega$. At the
point, we have
\begin{gather*}
\nabla \left (\ln(\underline{u}_0(x_0))-\ln ( \bar{u}_0(x_0))\right)=0,\\
\Delta \left (\ln(\underline{u}_0(x_0))-\ln(\bar{u}_0(x_0))\right)\leq 0.
\end{gather*}
By (G2), we see that
 \begin{align*}
&\Delta \left (\ln(\underline{u}_0(x_0))-\ln (\bar{u}_0(x_0))\right)\\
& = \frac {\Delta \underline{u}_0(x_0)}
 {\underline{u}_0(x_0)}- \frac {\Delta \bar{u}_0(x_0)}{\bar{u}_0(x_0)}
 -\frac {|\nabla
 \underline{u}_0(x_0)|^2}{(\underline{u}_0(x_0))^2}+\frac {|\nabla
 \bar{u}_0(x_0)|^2}{(\bar{u}_0(x_0))^2}\\
&=\frac {\Delta \underline{u}_0(x_0)}{\underline{u}_0(x_0)}-
 \frac {\Delta \bar{u}_0(x_0)}{\bar{u}_0(x_0)}\\
&\geq p(x_0)\Big(\big[\frac{G(\bar{u}_0(x_0))}{\bar{u}_0(x_0)}
  -\frac{G(\underline{u}_0(x_0))}{\underline{u}_0(x_0)}\big]
  +\frac{|\nabla \bar{u}_0(x_0)|^q}{\bar{u}_0(x_0)}\Big) > 0,
\end{align*}
which is a  contradiction. Therefore,
$\overline{u}_0\geq\underline{u}_0$ in $\Omega$. Then by  Lemma
\ref{lem2.1},  problem \eqref{e2.3} has at least one classical
solution denoted by $\bar{u}$, which is a super-solution
of problem \eqref{e2.1}.

Finally, we show that $\underline{u}\leq\bar{u}$ in $\Omega$.
Assume the contrary, i.e.,  there exists
 $x_1\in \Omega$ such that $\bar{u}(x_1)<\underline{u}(x_1)$. Then,
$\sup_{x\in \Omega} \left (\ln(\underline{u}(x))-\ln
(\bar{u}(x))\right )$ exists and is positive in $\Omega$. At the
point, we have
$$
\nabla \left (\ln(\underline{u}(x_1))-\ln ( \bar{u}(x_1))\right)=0
\quad\text{and}\quad
 \Delta \left (\ln(\underline{u}(x_1))-\ln(\bar{u}(x_1))\right)\leq 0.
$$
Since $\overline \sigma_0$  is non-increasing  on $(0,\infty)$, we
have
$$
\overline \sigma_0(\bar{u}(x_1))\geq\overline \sigma_0(\underline{u}(x_1))
\geq\frac{g(\underline{u}(x_1))+f(\underline{u}(x_1))}{\underline{u}(x_1)}.
$$
Then we obtain
 \begin{align*}
 &\Delta \left (\ln(\underline{u}(x_1))-\ln (\bar{u}(x_1))\right)\\
 & = \frac {\Delta \underline{u}(x_1)}
  {\underline{u}(x_1)}- \frac {\Delta \bar{u}(x_1)}{\bar{u}(x_1)}
 -\frac {|\nabla
 \underline{u}(x_1)|^2}{(\underline{u}(x_1))^2}+\frac {|\nabla
 \bar{u}(x_1)|^2}{(\bar{u}(x_1))^2}\\
&= \frac {\Delta \underline{u}(x_1)}{\underline{u}(x_1)}- \frac
 {\Delta \bar{u}(x_1)}{\bar{u}(x_1)}\\
&=  p(x_1)\Big(\big[\frac{G(\bar{u}(x_1))}{\bar{u}(x_1)}
  -\frac{g(\underline{u}(x_1))}{\underline{u}(x_1)}\big]
  +\frac{|\nabla \bar{u}(x_1)|^q}{\bar{u}(x_1)}\Big)\\
&=  p(x_1)\Big(\big[\overline \sigma_0(\bar{u}(x_1))
  -\frac{g(\underline{u}(x_1))}{\underline{u}(x_1)}\big]
  +\frac{1}{\bar{u}(x_1)}+\frac{|\nabla \bar{u}(x_1)|^q}{\bar{u}(x_1)}\Big)> 0,
\end{align*}
which is a  contradiction. Therefore,
$\overline{u}\geq\underline{u}$ in $\Omega$. By Lemma \ref{lem2.1},  the
proof is complete.
\end{proof}


\begin{lemma}\label{lem2.4}
 Assume  condition \eqref{e1.2} is fulfilled. Then there
is a function $w$ such that
\begin{equation}
\begin{gathered}
-\Delta w \geq p(x)[g(w)+f(w)+|\nabla w|^q], \quad w>0, \;  x\in \mathbb{R}^N,\\
w(x)\to 0,\quad\text{as } |x|\to \infty.
\end{gathered}\label{e2.20}
\end{equation}
\end{lemma}

\begin{proof} Denote
$$
\Psi(r):=r^{1-N}\int_0^rt^{N-1}\varphi(t)dt,\quad  \forall   r>0.
$$
By condition \eqref{e1.2} and the L'H\^{o}pital's rule, we have
$\lim_{r\to 0}\Psi(r)=\lim_{r\to \infty}\Psi(r)=0$.
Thus, $\Psi$ is bounded
 on $(0,\infty)$ and it can be extended in the origin by taking $\Psi(0)=0$.
On the other hand,
by integration by parts and the L'H\^{o}pital's rule (see details in
\cite{g2}) , we get
$$
\int_0^\infty\Psi(r)dr=\lim_{r\to \infty}\int_0^r\Psi(t)dt
=\frac{1}{N-2}\int_0^\infty r\varphi(r)dr<\infty.
$$
Let $\mu>2$ be such that
\begin{equation}
\mu^{1-q}\geq2\max_{r\geq0}\Psi^q(r).\label{e2.21}
\end{equation}
Define
$$
\rho(x):=\mu\int_{|x|}^{\infty}\Psi(t)dt, \quad \text{for } x\in \mathbb{R}^N.
$$
Then $\rho$ is bounded and  satisfies
\begin{gather*}
-\Delta \rho=\mu\varphi(|x|),\quad \rho>0,\; x\in\mathbb{R}^N,\\
\rho(x)\to 0, \quad \text{as }  |x|\to \infty.
\end{gather*}
We claim that there are $R>0$ and a function $w\in C^2(\mathbb{R}^N)$
 such that
\begin{equation}
\rho(x)=\frac{1}{R}\int_0^{w(x)}\frac{t}{G(t)+1}dt.\label{e2.22}
\end{equation}
Indeed, since
$$
\lim_{r\to  +\infty}\frac{\int_0^r\frac{t}{G(t)+1}dt}{r}
=\lim_{r\to  +\infty}\frac{r}{G(r)+1}=+\infty,
$$
we notice first for some $R>0$
$$
|\rho|_\infty\leq\frac{1}{R}\int_0^{R}\frac{t}{G(t)+1}dt,
$$
and in particular
\begin{equation}
w(x)\leq R, \ x\in \mathbb{R}^N.\label{e2.23}
\end{equation}
From \eqref{e2.22} we have
$$
|\nabla w|=R\frac{G(w)+1}{w}|\nabla \rho|=\mu R\Psi(|x|)\frac{G(w)+1}{w},
$$
and combining with (G2), we get
$$
\frac{1}{R}\frac{w}{G(w)+1}\Delta w+\frac{1}{R}\frac{d}{dw}
\left(\frac{w}{G(w)+1}\right)|\nabla w|^2=\Delta \rho,
$$
i.e.,
$$
-\Delta w  \geq \mu R\varphi(|x|)\frac{G(w)+1}{w}.
$$
Then  from  \eqref{e2.21}, \eqref{e2.23} and (G2), we obtain
 \begin{align*}
-\Delta w & \geq \mu R\varphi(|x|)\frac{G(w)+1}{w}\\
 &\geq  R p(x)\frac{G(w)+1}{w}+\frac{\mu}{2}R p(x)\frac{G(w)+1}{w}\\
 &\geq p(x)(G(w)+1)+p(x)|\mu R\frac{G(w)+1}{w}\Psi(|x|)|^q\\
 & = p(x)(G(w)+1)+p(x)|\nabla w|^q.
\end{align*}
Hence,
\begin{gather*}
-\Delta w \geq p(x)[G(w)+1+|\nabla w|^q],\  w>0, \  x\in \mathbb{R}^N,\\
w(x)\to 0, \ {\rm as }\quad |x|\to \infty.
\end{gather*} % \label{e2.24}
Since $G(w)>g(w)+f(w)$ on $(0,+\infty)$, it follows that $w$
satisfies \eqref{e2.20}. The proof is complete.
\end{proof}

\begin{proof}[Proof  of Theorem \ref{thm1.1}]
Consider the perturbed problem
\begin{equation}
-\Delta u_n =p(x)[g(u_n)+f(u_n)+|\nabla u_n|^q], \quad u_n>0, \; x\in B_n,
\;  u_n|_{\partial B_n}=0,
\label{e2.25}
\end{equation}
where $B_n:=\{x\in\mathbb{R}^N; |x|<n\}$, $n=1,2,3,\dots$.
It follows by Lemma \ref{lem2.3} that problem \eqref{e2.25} has at
least one solution $u_n\in C^2(B_n) \cap C(\bar{B_n})$.
Put
$$
u_n(x)=0, \quad  \forall\ |x|>n.
$$
Let  $w$ be as in Lemma \ref{lem2.4}, with the same proof  above,
 we deduce that
\begin{equation}
u_n(x)\leq w(x),\  x\in \mathbb{R}^N, \   n=1, 2, 3,
\dots.\label{e2.26}
\end{equation}
Now, we need to estimate $\{u_n\}$. For any bounded
$C^{2+\alpha}$-smooth domain $\Omega'\subset \mathbb{R}^N$, take
$\Omega_1$  and $\Omega_2$ with  $C^{2+\alpha}$-smooth boundaries,
and $K_1$ large enough, such that
$$
\Omega'\subset\subset \Omega_1\subset\subset
\Omega_2\subset\subset  B_n, \quad n\geq K_1.
$$
Note that
\begin{equation}
u_n(x)\geq \underline{u}(x)>0,\quad \forall  x\in B_{K_1}, \label{e2.27}
\end{equation}
where $B_{K_1}$  is the substitution for $\Omega$ in  the proof of
Lemma \ref{lem2.3}.
Let
$$
\Psi_n(x)=p(x)[g(u_n)+f(u_n)+|\nabla u_n|^q],\quad    x\in \bar{B}_{K_1}.
$$
Since $-\Delta u_n(x)=\Psi_n(x)$, $x\in B_{K_1}$, by the interior
estimate theorem of Ladyzenskaja and Ural'tseva
\cite[Theorem 3.1, p. 266]{l1}, we get a positive constant $C_1$ independent
of $n$ such that
$$
\max_{x\in\bar{\Omega}_2}|\nabla u_n(x)|
\leq C_1\max_{x\in \bar{B}_{K_1}}u_n(x)\leq C_1 \max_{x\in
\bar{B}_{K_1}}w(x),\quad \forall \ x\in B_{K_1},
$$ %\label{e2.28}
i.e., $|\nabla u_n(x)|$ is uniformly bounded on $\bar{\Omega}_2$.
It follows  that $\{\Psi_n\}_{K_1}^\infty$ is uniformly bounded on
$\bar{\Omega}_2$ and hence $\Psi_n\in L^p(\Omega_2)$ for any
$p>1$. Since $-\Delta u_n(x)=\Psi_n(x), \ x\in \Omega_2, $ we see
by \cite[Theorem 9.11]{g3},  that there exists a positive constant
$C_2$ independent of $n$ such that
$$
\|u_n\|_{W^{2, p}(\Omega_1)}\leq
C_2\left(\|\Psi_n\|_{L^p(\Omega_2)}+\|u_n\|_{L^p(\Omega_2)}
\right),\quad \forall\  n\geq K_1.
$$ % \label{e2.29}
Taking $p>N$ such that $\alpha< 1-N/p$  and applying Sobolev's
embedding inequality, we see that
$\{\|u_n\|_{C^{1+\alpha}(\bar{\Omega}_1)}\}_{K_1}^\infty$ is
uniformly bounded. Therefore $\Psi_n\in C^\alpha(\bar{\Omega}_1)$
and $\{\|\Psi_n\|_{C^\alpha(\bar{\Omega}_1)}\}_{K_1}^\infty$ is
uniformly bounded. It follows by Schauder's interior estimate
theorem (see \cite[Chapter 1, p. 2]{g3}) that  there exists a
positive constant $C_3$ independent of $n$ such that
$$
\|u_n\|_{C^{2+\alpha}(\bar{\Omega}')}\leq
C_3\left(\|\Psi_n\|_{C^\alpha(\bar{\Omega}_1)}+\|u_n\|_{C(\bar{\Omega}_1)}
\right),\quad \forall \ n\geq K_1;
$$ \label{e2.30}
 i.e., $\{\|u_n\|_{C^{2+\alpha}(\bar{\Omega}')}\}_{K_1}^\infty$
is uniformly bounded. Using Ascoli-Arzela's theorem and the
diagonal sequential process, we see that $\{u_n\}_{K_1}^\infty$
has a subsequence that converges uniformly in the
$C^2(\bar{\Omega}')$ norm to a function $u\in C^2(\bar{\Omega}')$
and $u$ satisfies
$$
- \Delta u=p(x)[g(u)+f(u)+|\nabla u|^q], \quad  x\in \bar{\Omega}'.
$$
By \eqref{e2.27}, we obtain that
$$
u>0, \quad \forall   x\in \bar{\Omega}'.
$$
Applying  Schauder's regularity theorem we see
that  $u\in C^{2+\alpha}(\bar{\Omega}')$. Since $\Omega'$ is
arbitrary, we also see that $u\in C^{2+\alpha}_{\rm loc}(\mathbb{R}^N)$.
 It follows by  \eqref{e2.26}  that $\lim_{|x|\to  \infty}u(x)=0$.
Thus, a standard bootstrap argument  shows that
 $u$ is a classical  solution
to problem \eqref{e1.1}. The proof is complete.
\end{proof}

At last,  it is worth  pointing out  that Ye and Zhou  \cite{y1} proved  that
 in many situations
  condition \eqref{e1.2}  can be replaced by the following more general
condition
\begin{itemize}
\item [(P1)] $-\Delta u=p(x)$ has a bounded ground state solution.
\end{itemize}
Obviously,  condition \eqref{e1.2}  implies (P1) (see \cite{y1} for
details about comparison between condition \eqref{e1.2}  and (P1)).
Therefore, we have an  unsolved problem as follows.

\begin{remark}
We note that the existence of ground
state solutions for problem \eqref{e1.1} is left an open problem if $p$
satisfies   condition (P1) instead of \eqref{e1.2}.
\end{remark}


\begin{thebibliography}{00}

\bibitem{a1} R. Agarwal, D. O'Regan;
\emph{Existence theory for single and multiple
solutions to singular positone boundary value problems}, J.
Differential Equations, {\bf 175} (2001), 393-414.

\bibitem {c1} F. C\v{i}rstea,  V. D. R\v{a}dulescu;
\emph{Existence and uniqueness
 of positive solutions to a semilinear elliptic problem
 in $\mathbb{R}^N$},  J. Math. Anal. Appl. {\bf  229} (1999),  417-425.

\bibitem {c2} F. C\v{i}rstea,  V. D. R\v{a}dulescu;
\emph{Combined effects of asymptotically
linear ans singular nonlinearities in bifurcation problems of
Lane-Emden-Fowler type},   J. Math. Pures Appl.
 {\bf  84} (2005),  493-508.


\bibitem{c3} S. Cui;
\emph{Existence and nonexistence of positive solutions for singular
semilinear elliptic boundary value problems}, Nonliear Anal. {\bf 41}
(2000), 149-176.

\bibitem {d1}  T. -L. Dinu;
\emph{Entire   positive solutions
of  the singular Emder-Fowler  equation with nonlinear gradient
term},   Results  Math. {\bf    43} (2003),  96-100.

\bibitem {d2} T. -L. Dinu;
\emph{Entire  solutions of  sublinear elliptic equations
in anisotropic media},  J. Math. Anal. Appl. {\bf 322} (2006), 382-392.

\bibitem{g1} M. Ghergu, V. R\v{a}dulescu;
\emph{On a class of sublinear singular
elliptic problems with convection term}, J. Math.Anal.Appl. {\bf  311}
(2005), 635-646.

\bibitem{g2} M. Ghergu, V. R\v{a}dulescu;
\emph{Ground state solutions for the  singular
 Lane-Emden-Fowler equation  with sublinear  convection term},
J. Math. Anal. Appl., in press.

\bibitem{g3} D. Gilbarg, N. S. Truginger;
\emph{Elliptic  Partial Differetial Equations
of Second Order}, 3rd ed., Springer, Berlin (1998).

\bibitem{g4} J. V. Goncalves, C. A. Santos;
 \emph{Existence  and asymptotic behavior of non-radially
  symmetric ground states of semilinear singular elliptic
   equations}, Nonlinear Anal. {\bf 65} (2006), 719-727.

\bibitem{g5} J. V. Goncalves, C. A. Santos;
\emph{Singular elliptic problems: Existence,
non-existence and boundary behavior}, Nonlinear Anal., {\bf 66}
(2007), 2078-2090

\bibitem{l1} O. A. Ladyzenskaja, N. N. Ural'tseva;
\emph{Linear and Quasilinear Elliptic
Equations}, Academic Press, 1968.

\bibitem {l2}  A. V. Lair,   A. W. Shaker;
 \emph{Entire  solutions of a singular elliptic
problem}, J. Math. Anal. Appl. {\bf 200} (1996), 498-505.

\bibitem{s1} J. Shi, M. Yao;
\emph{On a singular nonlinear semilinear elliptic problem},
Proc. Roy. Soc. Edinburgh Sect. A.  {\bf 128} (1998), 1389-1401.

\bibitem{s2} J. Shi, M. Yao; \emph{Positive solutions for elliptic equations
with singular nonlinearity}, Electronic of J. Differential
Equations,  2005 (2005), No. 04, 1-11.

\bibitem{s3} Y. Sun, S. Li;
\emph{Structure of ground state solutions of singular semilinear
elliptic equation}, Nonlinear Anal. {\bf  55} (2003), 399-417.

\bibitem{y1} D. Ye, F. Zhou;
\emph{Invariant criteria for existence of bounded positive
solutions}, Discrete and Continous Dynamical Systems  {\bf 12} (2005),
413-424.

\bibitem{z1} Z. Zhang;
\emph{Nonexistence of positive classical solutions of a singular
nonlinear Dirichlet Problem with a convection term}, Nonlinear Anal.
{\bf 27} (1996), 957-961.

\bibitem{z2} Z. Zhang;
\emph{A remark on the existence of entire solutions of a singular semilinear
 elliptic problem}, J. Math. Anal. Appl. {\bf 215} (1997), 579-582.

\bibitem{z3} Z. Zhang, J. Yu;
\emph{On a singular nonlinear Dirichlet problem with a convection term},
 SIAM J. Math. Anal. {\bf  32} (2000), 916-927.

\bibitem{z4} Z. Zhang;
\emph{The asymptotic behavior of the unique solution for the singular
Lane-Emden-Fowler equation}, J. Math. Anal. Appl. {\bf 312} (2005), 33-43.

\bibitem{z5} Z. Zhang;
\emph{A remark on the existence of positive entire solutions of a
sublinear elliptic problem},  Nonlinear Anal., in press.

\end{thebibliography}
\end{document}