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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 88, 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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/88\hfil Asymptotic behavior]
{Asymptotic behavior of ground state solutions for sublinear and
singular  nonlinear Dirichlet problems}

\author[Rym Chemmam, Abdelwaheb Dhifli, Habib M\^{a}agli\hfil EJDE-2011/88\hfilneg]
{Rym Chemmam, Abdelwaheb Dhifli, Habib M\^aagli} % in alphabetical order

\address{Rym Chemmam \newline
 D\'epartement de Math\'ematiques, Facult\'e des
Sciences de Tunis, Universit\'e Tunis El Manar,
Campus universitaire, 2092 Tunis, Tunisia}
\email{Rym.Chemmam@fst.rnu.tn}

\address{Abdelwaheb Dhifli \newline
D\'epartement de Math\'ematiques, Facult\'e des
Sciences de Tunis, Universit\'e Tunis El Manar,
Campus universitaire, 2092 Tunis, Tunisia}
\email{dhifli\_waheb@yahoo.fr}

\address{Habib M\^aagli \newline
 D\'epartement de Math\'ematiques, Facult\'e des
Sciences de Tunis, Universit\'e Tunis El Manar,
Campus universitaire, 2092 Tunis, Tunisia}
\email{habib.maagli@fst.rnu.tn}

\thanks{Submitted April 13, 2011. Published July 5, 2011.}
\subjclass[2000]{31B05, 31C35, 34B27, 60J50}
\keywords{Asymptotic behavior; Dirichlet problem; ground sate solution;
\hfill\break\indent singular equations; sublinear equations}

\begin{abstract}
 In this article, we are concerned with the asymptotic behavior
 of the classical solution to the semilinear boundary-value
 problem
 \[
 \Delta u+a(x)u^{\sigma }=0
 \]
 in $\mathbb{R}^n$, $u>0$, $\lim_{|x|\to \infty }u(x)=0$,
 where $\sigma <1$. The special feature is to consider the
 function $a$ in $C_{\rm loc}^{\alpha }(\mathbb{R}^n)$,
 $0<\alpha <1$, such that there exists $c>0$ satisfying
 \[
 \frac{1}{c}\frac{L(|x| +1)}{(1+|x| )^{\lambda }}
 \leq a(x)\leq c\frac{L(|x| +1)}{(1+|x| )^{\lambda }},
 \]
 where  $L(t):=\exp \big(\int_1^t\frac{z(s)}{s}ds\big)$,
 with $z\in C([1,\infty ))$ such that $\lim_{t\to \infty } z(t)=0$.
 The comparable asymptotic rate of $a(x)$ determines the asymptotic
  behavior of the solution.
\end{abstract}

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

\section{Introduction}

 In this article, we are interested in  estimates for positive
solutions of the  semilinear problem
\begin{equation} \label{e1}
\begin{gathered}
-\Delta u=a(x)g(u),\quad x\in\mathbb{R}^n,\;n\geq 3 \\
u>0\quad \text{in }\mathbb{R}^n, \\
\lim_{|x| \to \infty } u(x)=0.
\end{gathered}
\end{equation}
The existence of such solutions and their asymptotic behavior
have been extensively studied by many authors when \eqref{e1} has
a smooth bounded domain $\Omega $ with zero boundary
Dirichlet condition. We refer the reader to \cite{SON,cr,sy,ma,mz,20}
and the references therein.

 In recent years, the study of ground state solutions of problem \eqref{e1}
received a lot of interest and numerous existence results have been
established (see for instance \cite{br,ed,rad,ls,MO,san} and the
references therein).

More specifically, Lair and Shaker \cite{ls} established the existence
and the uniqueness of positive classical solution, where $g$ is
a positive nonincreasing and continuously differentiable function
on $(0,\infty )$ and $ a$ is a nontrivial nonnegative function
in $C_{\rm loc}^{\alpha }(\mathbb{R}^n)$, satisfying
\begin{equation}
\int_{0}^{\infty }t\max_{|x| =t} a(x)dt<\infty.  \label{1b}
\end{equation}
Moreover, they showed that this condition on $a$ is nearly optimal.

Furthermore,  Brezis and Kamin \cite{br} proved the existence of a
unique positive solution to the  problem
\begin{gather*}
-\Delta u=a(x)u^{\sigma },\quad x\in \mathbb{R}^n,\; n\geq 3 \\
u>0, \\
\liminf_{|x| \to \infty } u(x)=0,
\end{gather*}
where $0<\sigma <1$ and $a$ is a nonnegative measurable function
potentially bounded, that is the function
$x\mapsto \int_{\mathbb{R}^n}\frac{a(y)}{| x-y| ^{n-2}}dy$
is in $L^{\infty }(\mathbb{R}^n)$.

 Throughout this article,  we denote $\mathcal{K}$ the set of all
functions $L$ defined on $[1,\infty ) $, by
\[
L( t) :=c\exp \Big(\int_1^t\frac{z(s)}{s}ds\Big),
\]
where $c$ is a positive constant and $z\in C([1,\infty ) )$ such
that $\lim_{t\to \infty } z(t)=0$.

\begin{remark} \label{rmk1} \rm
It is obvious  that $L\in \mathcal{K}$ if and only if $L$ is a
positive function in $C^{1}( [1,\infty )) $ such that
$\lim_{t\to \infty } \frac{tL'(t)}{L(t)}=0$.
\end{remark}

\begin{example} \label{exmp1} \rm
Let $m\in \mathbb{N}^{\ast }$,
$( \lambda _1,\lambda _{2},\dots ,\lambda _{m}) \in {\mathbb{R}}^{m}$
and $\omega $ be a positive real number sufficiently large
such that the function
\[
L(t)=\prod_{k=1}^{m}( \log _{k}(wt)) ^{-\lambda _{k}}
\]
is defined and positive on $[1,\infty ) $, where $\log
_{k}x=\log \circ \log \circ \dots \circ \log x$ $($k times$).$Then $L\in
\mathcal{K}$.
\end{example}

In this paper, we give precise asymptotic behavior of the solution to
the problem
\begin{equation} \label{e2}
\begin{gathered}
-\Delta u=a(x)u^{\sigma },\quad x\in \mathbb{R}^n,\; n\geq 3, \\
u>0\quad\text{ in }\mathbb{R}^n, \\
\lim_{|x| \to \infty } u(x)=0,
\end{gathered}
\end{equation}
where $\sigma <1$ and  $a$ satisfies the hypothesis
\begin{itemize}
\item[(H1)]  $a$ is a nonnegative function in
$C_{\rm loc}^{\alpha}(\mathbb{R}^n)$, $0<\alpha <1$,
 satisfying
\[
a(x)\approx \frac{L(1+|x| )}{(1+|x| )^{\lambda }},
\]
where $\lambda \geq 2$ and $L\in \mathcal{K}$ such that
$\int_1^{\infty}t^{1-\lambda }L( t) dt<\infty $.
\end{itemize}
Here and throughout the paper, for two nonnegative functions
$f$ and $g$ defined on a set $S$, the notation
$f( x) \approx g(x)$, for $x\in S$ means that there exists
$c>0$ such that $\frac{1}{c}f(x)\leq g(x)\leq cf(x)$,
for all $x\in S$.

\begin{remark} \label{rmk2} \rm
(i) Note that we need to verify the condition $\int_1^{\infty
}t^{1-\lambda }L( t) dt<\infty $ in hypothesis (H1), only for $
\lambda =2$ (see Remark \ref{rmk3}).

(ii) It is obvious to see that if $a$ satisfies
hypothesis $(H1)$, then $a$ is potentially bounded and $a$
verifies \eqref{1b}. This implies from \cite{ls} and \cite{br},
that problem \eqref{e2} has a unique classical positive
solution in $C^{2,\alpha }(\mathbb{R}^n)$.
Thus it becomes interesting to know the asymptotic behavior of such
solution, as $t\to \infty$.
\end{remark}

Our main result is the following.

\begin{theorem} \label{thm1}
Assume {\rm (H1)}. Then the solution $u$ of problem \eqref{e2} satisfies
\begin{equation}
u(x)\approx \theta _{\lambda }(x),  \label{1.2}
\end{equation}
where $x\in\mathbb{R}^n$, and $\theta _{\lambda }$ is defined on
$\mathbb{R}^n$ by
\begin{equation}
\theta _{\lambda }(x):=\begin{cases}
\big( \int_{|x| +1}^{\infty }\frac{L(t)}{t}dt\big)
^{1/(1-\sigma)}, & \text{for }\lambda =2, \\
\frac{( L(1+|x| ) ^{1/(1-\sigma)}}{
(1+|x| )^{(\lambda -2)/(1-\sigma )}}, & \text{for }
2<\lambda <n-(n-2)\sigma , \\
\frac{1}{(1+|x| )^{n-2}}\big(
\int_1^{|x| +2}\frac{L(t)}{t}dt\big) ^{1/(1-\sigma)},
 &\text{for }\lambda =n-(n-2)\sigma , \\
\frac{1}{(1+|x| )^{n-2}}, & \text{for }\lambda>n-(n-2)\sigma .
\end{cases}   \label{1.3}
\end{equation}
\end{theorem}

To obtain estimates \eqref{1.3}, we shall adopt a
sub-supersolution method. For the reader's convenience, we recall the
definition.

A positive function $v\in C^{2,\alpha }(\mathbb{R}^n)$ is called a
 subsolution of problem \eqref{e2} if
\begin{equation}
\begin{gathered}
-\Delta v\leq a(x)v^{\sigma }\quad x\in\mathbb{R}^n, \\
\lim_{|x| \to \infty } v(x)=0.
\end{gathered}   \label{1.4}
\end{equation}
If the above inequality is reversed, $v$ is called a supersolution of
problem \eqref{e2}.


 The outline of this article is as follows.
In Section 2, we state some already known results on functions
in $\mathcal{K}$, useful for our study
and we give estimates on some potential functions.
The proof of Theorem \ref{thm1} is given in Section 3.
The last section is reserved to some applications.

We close this section by giving the following notation.
For a nonnegative measurable function $a$ in
$\mathbb{R}^n$, we denote by $Va$ the potential of $a$ defined on
$\mathbb{R}^n$ by
\[
Va(x)=\int_{\mathbb{R}^n}G(x,y)a(y)dy,
\]
where $G(x,y)=\frac{c_{n}}{|x-y| ^{n-2}}$ is the Green
function of the Laplacian $\Delta $ in $\mathbb{R}^n$
$( n\geq 3)$, and
$c_{n}=\frac{\Gamma (\frac{n}{2}-1)}{4\pi ^{\frac{n}{2}}}$.
We point out that for any nonnegative function $f$ in
$C_{\rm loc}^{\alpha }(\mathbb{R}^n)$
$(0<\alpha <1)$ such that $Vf\in L^{\infty }(\mathbb{R}^n)$,
we have $Vf\in C_{\rm loc}^{2,\alpha }(\mathbb{R}^n)$ and
satisfies $-\Delta ( Vf) =f$ in $\mathbb{R}^n$;
see \cite[Theorem 6.3]{SC}.

\section{Key estimates}

\subsection{Technical lemmas}
In what follows, we collect some fundamental properties of functions
belonging to the class $\mathcal{K}$. First, we need the following
elementary result.

\begin{lemma}[Karamata's Theorem] \label{lem1}
Assume that $g\in C^{1}([\beta ,\infty ),(0,\infty))$ and that
$\lim_{t\to \infty }tg'(t)/g(t)=\gamma $. Then we have
the following properties:
\begin{itemize}
\item[(i)] If $\gamma <-1$, then $\int_{\beta }^{\infty }g(s)ds$
converges and
\[
\int_{t}^{\infty }g(s)ds \sim -\frac{tg(t)}{\gamma +1},
 \quad\text{as }t\to\infty .
\]

\item[(ii)] If $\gamma >-1$, then $\int_{\beta }^{\infty }g(s)ds$
diverges and
\[
\int_{\beta }^tg(s)ds \sim \frac{tg(t)}{\gamma +1}
\quad\text{as } t\to \infty.
\]
\end{itemize}
\end{lemma}

\begin{remark} \label{rmk3} \rm
Let $\gamma \in \mathbb{R}$ and $L$ be a function in $\mathcal{K}$.
Applying Lemma \ref{lem1} to $g(t)=t^{\gamma }L(t)$, we obtain that
\begin{itemize}
\item If $\gamma <-1$, then $\int_1^{\infty }s^{\gamma }L(s)ds$
diverges and $\int_{t}^{\infty }s^{\gamma }L(s)ds
\sim -\frac{t^{1+\gamma }L(t)}{\gamma +1}$, as $t\to \infty$;

\item  If $\gamma >-1$, then $\int_1^{\infty }s^{\gamma }L(s)ds$
converges and
$\int_1^ts^{\gamma }L(s)ds \sim \frac{t^{1+\gamma }L(t)}{\gamma +1}$
as $t\to \infty$.
\end{itemize}
\end{remark}

\begin{lemma} \label{lem2}
$(i)$ Let $L_1$, $L_{2}\in \mathcal{K}$, $p\in
\mathbb{R}$. Then  $L_1L_{2}\in \mathcal{K}$ and $L_1^{p}\in \mathcal{K}$.

$(ii)$ Let $L$ be a function in $\mathcal{K}$ then there exists
$m\geq 0$ such that for every $\eta >0$ and $t\geq 1$ ,we have
\[
( 1+\eta ) ^{-m}L( t) \leq L( \eta +t) \leq ( 1+\eta ) ^{m}L( t) .
\]
\end{lemma}

\begin{proof}
Assertion (i) is due to Remark \ref{rmk1}. Let us prove (ii). Let $z$ be the
function in $C([1,\infty ) )$ such that
$\lim_{t\to\infty } z(t)=0$ and
$L( t) =\exp \big(\int_1^t\frac{z(s)}{s}ds\big)$.

Put $m=\sup_{t\in [ 1,\infty )} | z(t)| $,
then for each $\eta >0$ and $t\geq 1$, we have
\[
m \log\frac{t}{t+\eta }\leq \int_{t}^{t+\eta }\frac{z(s)}{s}ds
\leq m\log\frac{t+\eta }{t}.
\]
That is,
\[
( 1+\frac{\eta }{t}) ^{-m}\leq \exp \Big( \int_{t}^{t+\eta }
\frac{z(s)}{s}ds\Big) \leq ( 1+\frac{\eta }{t}) ^{m}.
\]
So (ii) holds.
\end{proof}

\begin{lemma} \label{lem3}
Let $L\in \mathcal{K}$ and $\varepsilon >0$, then we have
\begin{gather}
\lim_{t\to \infty } t^{-\varepsilon }L(t)=0,  \label{2.1}\\
\lim_{t\to \infty } \frac{L( t) }{\int_1^t L(s)/s \,ds}=0.  \label{2.2}
\end{gather}
If further $\int_1^{\infty }L(s)/s\,ds$ converges, then
\begin{equation}
\lim_{t\to \infty } \frac{L( t) }{
\int_{t}^{\infty }L(s)/s \,ds}=0.  \label{2.3}
\end{equation}
\end{lemma}

\begin{proof}
Let $L\in \mathcal{K}$ and $\varepsilon >0$. It is obvious
by Remark \ref{rmk1} that the function $t\to t^{-\frac{\varepsilon }{2}}L(t)$ is
non-increasing in $[\omega ,\infty )$, for $\omega $ large enough.
Then
\[
t^{-\frac{\varepsilon }{2}}L(t)\leq \omega ^{-\frac{\varepsilon }{2}
}L(\omega ),\text{ for }t\geq \omega ;
\]
That is,
\[
t^{-\varepsilon }L(t)\leq \frac{L( \omega ) }{( \omega
t) ^{\varepsilon /2}},\quad \text{for }t\geq \omega .
\]
This proves \eqref{2.1}.
For the rest of the proof, we distinguish two cases.

\textbf{Case 1:}  $\int_1^{\infty }\frac{L(s)}{s}ds<\infty $.
Since the function $t\to tL(t)$ is nondecreasing in
$[\omega,\infty )$, then
\[
tL(t)\int_{t}^{\infty }\frac{ds}{s^{2}}\leq
\int_{t}^{\infty }\frac{L(s)}{s} ds,\quad \text{for }t\geq \omega .
\]
Hence
\[
0<L(t)\leq \int_{t}^{\infty }\frac{L(s)}{s}ds,\quad
\text{for }t\geq \omega .
\]
Then $\lim_{t\to \infty } L(t)=0$, which implies \eqref{2.2}.

Moreover, put $\varphi ( t) =L(t)/t$, for $t\geq 1$.
Since $\varphi $ satisfies
$\lim_{t\to \infty } t\varphi '(t)/\varphi(t)=-1$, then it follows
that
\[
\int_{t}^{\infty }\varphi ( s) ds
\sim -\int_{t}^{\infty }s\varphi '(s)ds=t\varphi
(t)+\int_{t}^{\infty }\varphi (s)ds,
\]
as $t\to \infty$.
This implies that
\[
\int_{t}^{\infty }\frac{L(s)}{s}ds \sim
L( t) +\int_{t}^{\infty }\frac{L(s)}{s}ds,
\]
as $t\to \infty$. So we deduce \eqref{2.3}.

\textbf{Case 2:}  $\int_1^{\infty }\frac{L(s)}{s}ds=\infty $.
Put $\varphi ( t) =L(t)/t$, for $t\geq 1$. Then for $
\omega $ sufficiently large and $t\geq \omega $, we have
\[
\int_{\omega }^t\varphi ( s) ds
\sim -t\varphi (t)+\omega \varphi ( \omega ) +\int_{\omega
}^t\varphi (s)ds,
\]
as $t\to \infty$; that is,
\[
\int_{\omega }^t\frac{L(s)}{s}ds \sim
-L(t)+\omega \varphi ( \omega ) +\int_{\omega }^t\frac{L(s)}{s}
ds,
\]
as $t\to \infty$.
Which proves \eqref{2.2} and completes the proof.
\end{proof}

\begin{remark} \label{rmk4} \rm
Let $L\in \mathcal{K}$. Using Remark \ref{rmk1} and \eqref{2.2}, we
deduce that
\[
t\to \int_1^{t+1}\frac{L(s)}{s}ds\in \mathcal{K}.
\]
If further $\int_1^{\infty } L(s)/s\,ds$ converges, we have by
\eqref{2.3} that
\[
t\mapsto \int_{t}^{\infty } \frac{L(s)}{s}\,ds\in \mathcal{K}.
\]
\end{remark}

\subsection{Asymptotic behavior of some potential functions}

We are going to give estimates on the potential functions $Va$ and
$V(a\theta _{\lambda }^{\sigma })$, where $\theta _{\lambda }$ is the
function given in \eqref{1.3}.

\begin{proposition} \label{prop1}
Let $a$ be a function satisfying {\rm (H1)}. Then for
$x\in\mathbb{R}^n$,
\[
Va( x) \approx \psi ( |x| ) ,
\]
where $\psi $ is the function defined in $[0,\infty )$ by
\begin{equation}
\psi (t)=\begin{cases}
\int_{t+1}^{\infty }L(r)/r\,dr  , &\text{for }\lambda =2, \\[3pt]
\frac{L(1+t)}{( 1+t) ^{\lambda -2}} ,& \text{for }2<\lambda <n,\\[3pt]
\frac{1}{( 1+t) ^{n-2}}\int_1^{t+2} L(r)/ r\, dr, &
\text{for }\lambda =n, \\[3pt]
\frac{1}{( 1+t) ^{n-2}}, & \text{for } \lambda >n.
\end{cases}  \label{2.3'}
\end{equation}
\end{proposition}

\begin{proof}
First, we recall the following well known result.
Let $\varphi $ be a nonnegative radial measurable function
and $x\in \mathbb{R}^n$, then we have
\[
\int_{\mathbb{R}^n}\frac{\varphi ( y) }{| x-y| ^{n-2}} dy
=c\int_{0}^{\infty }\frac{r^{n-1}}{\max (|x| ,r)^{n-2}}
\varphi ( r) dr.
\]
Now, let $\lambda \geq 2$ and $L\in \mathcal{K}$ satisfying
$\int_1^{\infty }t^{1-\lambda }L( t) dt<\infty $ and such that
\[
a(x)\approx \frac{L(1+|x| )}{(1+|x|)^{\lambda }}.
\]
Thus
\[
Va(x)\approx \int_{\mathbb{R}^n}\frac{L(1+| y| )}{(1+|
y| )^{\lambda }}\frac{1}{| x-y| ^{n-2}}
dy=c_{n}I(|x| ),
\]
where $I$ is the function defined on $[0,\infty ) $ by
\[
I(t)=\int_{0}^{\infty }\frac{r^{n-1}L(1+r)}{\max (t,r)^{n-2}
(1+r)^{\lambda }} dr.
\]
So to prove the result, it is sufficient to show that
$I( t) \approx \psi ( t) $ for $t\in [0,\infty ) $.
We have
\begin{align*}
I(t)
&=\frac{1}{t^{n-2}}\int_{0}^{1}\frac{r^{n-1}L(1+r)}{(1+r)^{\lambda }}
dr+\frac{1}{t^{n-2}}\int_1^t\frac{r^{n-1}L(1+r)}{(1+r)^{\lambda }}
dr+\int_{t}^{\infty }\frac{rL(1+r)}{(1+r)^{\lambda }}dr \\
&:=I_1(t)+I_{2}(t)+I_{3}(t).
\end{align*}
It is clear that for $t\geq 2$,
\begin{equation}
I_1(t)\approx \frac{1}{t^{n-2}}.  \label{2.4}
\end{equation}
To estimate $I_{2}$ and $I_{3}$, we distinguish two cases.

\textbf{Case 1:} $\lambda >2$.
Using Lemma \ref{lem2} (ii) and Remark \ref{rmk3},  for $t\geq 2$ we have
\begin{equation}
I_{3}(t)\approx \int_{t}^{\infty }r^{1-\lambda }L(r)dr
\approx \frac{L(t)}{t^{\lambda -2}}.  \label{2.5}
\end{equation}
$\bullet $ If $2<\lambda <n$, then applying again Remark \ref{rmk3}, we have
$\int_1^{\infty }r^{n-1-\lambda }L(r)dr=\infty $ and
$\int_1^tr^{n-1-\lambda }L(r)dr \sim t^{2-\lambda }L(t)$,
as $t\to \infty$. So
by Lemma \ref{lem2} (ii), for $t\geq 2$ we obtain
\[
I_{2}(t)\approx \frac{1}{t^{n-2}}\int_1^tr^{n-1-\lambda }L(r)dr\approx
\frac{L(t)}{t^{\lambda -2}}.
\]
Then by \eqref{2.4}, \eqref{2.5} and \eqref{2.1},  for $t\geq 2$ we have
\[
I(t)\approx \frac{1}{t^{n-2}}+\frac{L(t)}{t^{\lambda -2}}\approx \frac{L(t)}{
t^{\lambda -2}}.
\]
Now, since the functions $t\to I( t) $ and
$t\to \frac{L(1+t)}{( 1+t) ^{\lambda -2}}$ are positive and continuous
in $[0,\infty )$,  for $t\geq 0$ we obtain
\[
I(t)\approx \frac{L(1+t)}{( 1+t) ^{\lambda -2}}.
\]
$\bullet $ If $\lambda >n$, then applying Remark \ref{rmk3}, we have
$\int_1^tr^{n-1-\lambda }L(r)dr<\infty $. So by Lemma \ref{lem2} (ii),
for $t\geq 2$, we obtain
\[
I_{2}(t)\approx \frac{1}{t^{n-2}}\int_1^tr^{n-1-\lambda }L(r)dr\approx
\frac{1}{t^{n-2}}.
\]
This together with \eqref{2.4}, \eqref{2.5} and \eqref{2.1} implies
that for $t\geq 2$,
\[
I(t)\approx \frac{1}{t^{n-2}}.
\]
Then by the same argument as above, we deduce that for $t\geq 0$,
\[
I(t)\approx \frac{1}{( 1+t) ^{n-2}}.
\]
$\bullet $ If $\lambda =n$, then using \eqref{2.4}, \eqref{2.5}
and \eqref{2.2},  for $t\geq 2$, we have
\[
I(t) \approx \frac{1}{t^{n-2}}( 1+\int_1^t\frac{L(r)}{r}
dr+L(t))
\approx \frac{1}{t^{n-2}}\int_1^t\frac{L(r)}{r}dr.
\]
So  for $t\geq 0$, we obtain
\[
I(t)\approx \frac{1}{( 1+t) ^{n-2}}\int_1^{t+2}\frac{L(r)}{r}
dr.
\]

\textbf{Case 2:} $\lambda =2$.
By Remark \ref{rmk3},  for $t\geq 2$, we have
$I_{2}(t)\approx L(t)$.
So  for $t\geq 2$, we have
\[
I(t)\approx \frac{1}{t^{n-2}}+L(t)+\int_{t}^{\infty }\frac{L(r)}{r}dr.
\]
Hence using \eqref{2.1} and \eqref{2.3},  for $t\geq 2$, we have
\[
I(t)\approx \int_{t}^{\infty }\frac{L(r)}{r}dr.
\]
So for $t\geq 0$, we obtain
\[
I(t)\approx \int_{t+1}^{\infty }\frac{L(r)}{r}dr.
\]
This completes the proof.
\end{proof}

The following Proposition plays a key role in this paper.

\begin{proposition} \label{prop2}
Let $a$ be a function satisfying (H1) and let
$\theta_{\lambda }$ be the function given by \eqref{1.3}.
Then for $x\in\mathbb{R}^n$,
\[
V(a\theta _{\lambda }^{\sigma })(x)\approx \theta _{\lambda }(x).
\]
\end{proposition}

\begin{proof}
Let $\lambda \geq 2$ and $L\in \mathcal{K}$ satisfying
$\int_1^{\infty}t^{1-\lambda }L( t) dt<\infty $ and such that
\[
a(x)\approx \frac{L(1+|x| )}{(1+|x|
)^{\lambda }}.
\]
Then for every $x\in \mathbb{R}^n$, we have
\[
a(x)\theta _{\lambda }^{\sigma }(x)\approx h(x)
:=\begin{cases}
\frac{L(1+|x| )}{(1+|x| )^{2}}
\big( \int_{|x| +1}^{\infty }\frac{L(t)}{t}dt\big)
^{\sigma/(1-\sigma)}, &  \lambda =2, \\[3pt]
\frac{( L(1+|x| ) ^{1/(1-\sigma)}}{
(1+|x| )^{(\lambda -2\sigma)/(1-\sigma )}}, &
 2<\lambda <n-(n-2)\sigma , \\[3pt]
\frac{L(1+|x| )}{(1+|x| )^n}
\big( \int_1^{|x| +2}\frac{L(t)}{t}dt\big) ^{\sigma/(1-\sigma)},
&\lambda =n-(n-2)\sigma , \\[3pt]
\frac{L(1+|x| )}{(1+|x| )^{\lambda
+(n-2)\sigma }}, & \lambda >n-(n-2)\sigma .
\end{cases}
\]

We point out that
$h(x)=\frac{\widetilde{L}(1+|x| )}{(1 +|x| )^{\mu }}$,
where $\mu \geq 2$.
Moreover, due to Lemma \ref{lem2} and Remark \ref{rmk4}, we deduce that
$\widetilde{L}\in \mathcal{K}$ and satisfies
$\int_1^{\infty }t^{1-\mu }\widetilde{L}(t) dt<\infty $.
Hence, it follows by Proposition \ref{prop1}, that
\[
V(a\theta _{\lambda }^{\sigma })( x) \approx V(h)( x)
\approx \widetilde{\psi }( |x| ),\quad \text{in }
\mathbb{R}^n,
\]
where $\widetilde{\psi }$ is the function defined by \eqref{2.3'}
 by replacing $L$ by $\widetilde{L}$ and $\lambda $ by $\mu $.
This completes the proof by a simple calculus.
\end{proof}

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

Let $a$ be a function satisfying (H1). The main idea is to
find a subsolution and a supersolution of problem \eqref{e2}
of the form $cV(a\phi ^{\sigma })$, where $c>0$ and
$\phi (x)=\frac{L_{0}(1+|x| )}{(1+|x|) ^{\beta }}$, which
will satisfy necessarily
\begin{equation}
V(a\phi ^{\sigma })\approx \phi .  \label{3.1}
\end{equation}
So, the choice of the real $\beta $ and the function $L_{0}$
in $\mathcal{K}$ is such that \eqref{3.1} is satisfied.
Setting $\phi (x)=\theta _{\lambda}(x)$, where $\theta _{\lambda }$
is the function given by \eqref{1.3}, we have by Proposition \ref{prop2}
 that the function $\theta _{\lambda }$ satisfies \eqref{3.1}.

Let $v:=V(a\theta _{\lambda }^{\sigma })$ and let $M>1$ be such that
\[
\frac{1}{M}v\leq \theta _{\lambda }\leq Mv.
\]
Which implies that for $\sigma <1$,
\[
\frac{v^{\sigma }}{M^{| \sigma | }}\leq \theta _{\lambda
}^{\sigma }\leq M^{| \sigma | }v^{\sigma }.
\]
Put $c:=M^{|\sigma |/(1-\sigma)}$, then it is
easy to verify that $\underline{u}=\frac{1}{c}v$ and
$\overline{u}=cv$ are respectively a subsolution and a supersolution
of problem \eqref{e2}.

Now, since $c>1$, we get $\underline{u}\leq \overline{u}$ on
$\mathbb{R}^n$ and thanks to the method of sub and supersolution,
it follows that problem \eqref{e2} has a solution $u$ satisfying
$\underline{u}\leq u\leq \overline{u}$, in $\mathbb{R}^n$.

Finally, by using Remark \ref{rmk2} (ii) and
Proposition \ref{prop2}, we deduce
that the unique classical positive solution of problem \eqref{e2}
satisfies \eqref{1.2}. This completes the proof.

\section{Applications}

\subsection{First application}
Let $\sigma <1$ and $a$ be a positive function in
$C_{\rm loc}^{\alpha }(\mathbb{R}^n)$ satisfying for
$x\in\mathbb{R}^n$
\[
a(x)\approx \frac{1}{(1+|x| )^{\lambda }}
\prod_{k=1}^{m}( \log _{k}(w(1+|x| ))) ^{-\lambda _{\mathbf{k}}},
\]
where $m\in\mathbb{N}^{\ast }$ and $w$ is a positive constant
large enough. The real numbers $\lambda $ and $\lambda _{k}$,
$1\leq k\leq m$, satisfy one of the following two
conditions

$\bullet $ $\lambda >2$ and $\lambda _{k}\in
\mathbb{R}$ for $1\leq k\leq m$.

$\bullet $ $\lambda =2$ and $\lambda _1=\lambda _{2}=\dots =\lambda
_{l-1}=1$, $\lambda _{l}>1$, $\lambda _{k}\in \mathbb{R}$
for $l+1\leq k\leq m$.

Using Theorem \ref{thm1}, we deduce that problem \eqref{e2} has a unique
classical positive solution $u$ in
 $\mathbb{R}^n$ satisfying
\begin{itemize}
\item[(i)] If $\lambda =2$, then for
$x\in\mathbb{R}^n$
\[
u(x)\approx ( \log _{l}w(1+|x| )) ^{(1-\lambda _{l})/(1-\sigma)}
\prod_{k=l+1}^{m}( \log _{k}w(1+|x| )) ^{-\lambda _{k}/(1-\sigma)}.
\]

\item[(ii)] If $2<\lambda <n-\sigma (n-2)$, then for
$x\in\mathbb{R}^n$
\[
u(x)\approx \frac{1}{(1+|x| )^{(\lambda -2)/(1-\sigma)}}
\prod_{k=1}^{m}( \log _{k}w(1+|x| ))
^{-\lambda _{k}/(1-\sigma)}.
\]

\item[(iii)] If $\lambda =n-\sigma (n-2)$ and $\lambda _1=\lambda
_{2}=\dots =\lambda _{m}=1$, then for $x\in \mathbb{R}^n$
\[
u(x)\approx \frac{1}{(1+|x| )^{n-2}}( \log
_{m+1}w(1+|x| )) ^{1/(1-\sigma)}.
\]

\item[(iv)] If $\lambda =n-\sigma (n-2)$ and $\lambda _1=\lambda
_{2}=\dots =\lambda _{l-1}=1$, $\lambda _{l}<1$, $\lambda _{k}\in
\mathbb{R}$, for $l+1\leq k\leq m$, then for
$x\in \mathbb{R}^n$
\[
u(x)\approx \frac{1}{(1+|x| )^{n-2}}( \log
_{l}w(1+|x| )) ^{(1-\lambda _{l})/(1-\sigma )
}\prod_{k=l+1}^{m}( \log _{k}w(1+|x|
)) ^{-\lambda _{k}/(1-\sigma)}.
\]

\item[(v)] If $\lambda =n-\sigma (n-2)$ and $\lambda _1=\lambda
_{2}=\dots =\lambda _{l-1}=1$, $\lambda _{l}>1$, $\lambda _{k}\in
\mathbb{R}
$, for $l+1\leq k\leq m$, then for $x\in
\mathbb{R}
^n$
\[
u(x)\approx \frac{1}{(1+|x| )^{n-2}}.
\]

\item[(vi)] If $\lambda >n+\sigma (n-2)$, then for $x\in
\mathbb{R} ^n$
\[
u(x)\approx \frac{1}{(1+|x| )^{n-2}}.
\]
\end{itemize}

\subsection{Second application}

Let $a$ be a function satisfying (H1)  and let
$\sigma ,\beta <1$. We are interested in the  problem
\begin{equation}
\begin{gathered}
-\Delta u+\frac{\beta }{u}| \nabla u| ^{2}=a(x)u^{\sigma
}\quad \text{in }\mathbb{R}^n, \\
u>0,\quad \text{in }\mathbb{R}^n,\\
 \lim_{|x| \to \infty } u(x)=0.
\end{gathered} \label{4.1}
\end{equation}
Put $v=u^{1-\beta }$, then by a simple calculus, we obtain
that $v$ satisfies
\begin{equation}
\begin{gathered}
-\Delta v=( 1-\beta ) a(x)v^{\frac{\sigma -\beta }{1-\beta }} \quad
\text{in } \mathbb{R}^n, \\
v>0\quad \text{in } \mathbb{R}^n,\\
\lim_{|x| \to \infty } v(x)=0.
\end{gathered}   \label{4.2}
\end{equation}
Applying Theorem \ref{thm1} to problem \eqref{4.2}, we obtain that
there exists a unique solution $v$ such that
\[
v(x)\approx \widetilde{\theta }_{\lambda }( x)
:=\begin{cases}
\big( \int_{|x| +1}^{\infty }\frac{L(s)}{s}ds\big)
^{(1-\beta)/(1-\sigma)}, & \text{if }\lambda =2, \\[3pt]
\frac{( L(1+|x| )) ^{(1-\beta)/(1-\sigma)}}
{(1+|x| )^{(\lambda -2)/(1-\sigma)}}, &
 \text{if }2<\lambda <n-(n-2)\frac{\sigma -\beta }{1-\beta }, \\[3pt]
\frac{1}{(1+|x| )^{n-2}}\big(\int_1^{|x| +2}\frac{L(s)}{s}ds\big)
 ^{\frac{1-\beta}{1-\sigma}}, & \text{if }\lambda
 =n-(n-2)\frac{\sigma -\beta }{1-\beta }, \\[3pt]
\frac{1}{(1+|x| )^{n-2}}, & \text{if }\lambda >n-(n-2)
\frac{\sigma -\beta }{1-\beta }.
\end{cases}
\]
Consequently, we deduce that  \eqref{4.1} has a unique positive
solution $u$ satisfying
\begin{align*}
u(x)&\approx \big( \widetilde{\theta }_{\lambda }(x)\big)
 ^{1/(1-\beta)}\\
&=\begin{cases}
\Big( \int_{|x| +1}^{\infty }\frac{L(s)}{s}ds\Big)
^{1/(1-\sigma)}, & \text{if }\lambda =2, \\[3pt]
(1+|x| )^{\frac{2-\lambda}{(1-\sigma )( 1-\beta )}} ( L(1+|x| )) ^{1/(1-\sigma)},
 & \text{if }2<\lambda <n-(n-2)\frac{\sigma -\beta }{1-\beta
}, \\[3pt]
(1+|x| )^{(2-n)/(1-\beta)}\Big(
\int_1^{|x| +2}\frac{L(s)}{s}ds\Big) ^{1/(1-\sigma)},
& \text{if }\lambda =n-(n-2)\frac{\sigma -\beta }{1-\beta },
\\[3pt]
(1+|x| )^{(2-n)/(1-\beta)}, & \text{if }\lambda
>n-(n-2)\frac{\sigma -\beta }{1-\beta }.
\end{cases}
\end{align*}

\subsection{Third application}

Let $a$ be a function satisfying (H1)  and $L$ be a function
in $\mathcal{K}$ such that
\[
a(x)\approx \frac{L(1+|x| )}{(1+|x|)^{\lambda }}.
\]
Let $b\in C_{\rm loc}^{\gamma }(\mathbb{R}^n) $, $0<\gamma <1$
satisfying for $x\in \mathbb{R}^n$,
\[
b(x)\approx \frac{L_1(1+|x| )}{(1+|x| )^{\mu }},
\]
where $\mu \in \mathbb{R}$ and $L_1\in \mathcal{K}$.
Let $\sigma ,\beta <1$ and $p\in \mathbb{R}$. We are interested
in the  system
\begin{equation}
\begin{gathered}
-\Delta u=a(x)u^{\sigma }\quad \text{in } \mathbb{R}^n, \\
-\Delta v=b(x)u^{p}v^{\beta }\quad \text{in }\mathbb{R}^n, \\
u,v>0 \quad \text{in }\mathbb{R}^n,\
\lim_{|x| \to \infty }u(x)=\lim_{|x| \to \infty }v(x)=0.
\end{gathered}  \label{4.3}
\end{equation}
By Theorem \ref{thm1}, it follows that there exists a unique classical
solution $u$ to  \eqref{e2} satisfying \eqref{1.2}.
 So, we distinguish the following cases.

\textbf{Case 1:} $\lambda =2$.
By hypothesis (H1), we have $\int_1^{\infty }\frac{L(t)}{t}
dt<\infty $ and using estimates \eqref{1.3}, we deduce that
\[
b(x)u^{p}(x)\approx \frac{L_1(1+|x| )}{(1+|x| )^{\mu }}
\Big( \int_{|x| +1}^{\infty }
\frac{L(s)}{s}ds\Big) ^{p/(1-\sigma)}
:=\frac{L_{2}(1+|x| )}{(1+|x| )^{\mu }}.
\]
It is obvious to see by Lemma \ref{lem2} and Remark \ref{rmk4} that
$L_{2}\in \mathcal{K}$.
Now suppose that $\mu \geq 2$ and
$\int_1^{\infty }t^{1-\mu}L_{2}(t)dt<\infty $.
Then applying Theorem \ref{thm1}, we conclude that \eqref{4.3}
 has a unique classical solution $( u,v) $ such that
$u(x)\approx \theta _{\lambda }( x)$
and
\[
v(x)\approx \begin{cases}
\big( \int_{|x| +1}^{\infty }\frac{L_{2}(s)}{s}
ds\big) ^{1/(1-\beta )}, & \text{if }\mu =2, \\[3pt]
\frac{( L_{2}(1+|x| )) ^{1/(1-\beta )}}{(1+|x| )
^{(\mu -2)/(1-\beta)}}, & \text{if }
2<\mu <n-(n-2)\beta , \\[3pt]
\frac{1}{(1+|x| )^{n-2}}
\big( \int_1^{|x| +2}\frac{L_{2}(s)}{s}ds\big) ^{1/(1-\beta )},
& \text{if }\mu =n-(n-2)\beta , \\[3pt]
\frac{1}{(1+|x| )^{n-2}}, & \text{if }\mu >n-(n-2)\beta .
\end{cases}
\]


\textbf{Case 2:} $2<\lambda <n-(n-2)\sigma $.
Put $\gamma =\mu +\frac{\lambda -2}{1-\sigma }p$.
From the estimates \eqref{1.3}, we deduce that
\[
b(x)u^{p}( x) \approx \frac{L_1(1+|x|) ( L(1+|x| ))
^{p/(1-\sigma)}}
{(1+|x| )^{\gamma }}:=\frac{L_{2}(1+|x| )}{(1+|x| )^{\gamma }}.
\]
Obviously by Lemma \ref{lem2} we have that $L_{2}\in \mathcal{K}$.
 Now suppose that $\gamma \geq 2$ and
$\int_1^{\infty }t^{1-\gamma }L_{2}(t)dt<\infty $. Then
applying Theorem \ref{thm1}, we conclude that system \eqref{4.3} has
a unique classical solution $(u,v)$ such that
$u(x)\approx \theta _{\lambda }(x)$
and
\[
v(x)\approx \begin{cases}
\big(\int_{|x| +1}^{\infty }\frac{L_{2}(s)}{s}
ds\big) ^{1/(1-\beta )}, & \text{if }\gamma =2, \\[3pt]
\frac{L_{2}((1+|x| ))^{1/(1-\beta )}}{
(1+|x| )^{(\gamma -2)/(1-\beta)}}, & \text{if }
2<\gamma <n-(n-2)\beta , \\[3pt]
\frac{1}{(1+|x| )^{n-2}}\big( \int_1^{|x| +2}\frac{L_{2}(s)}{s}ds
\big) ^{1/(1-\beta )}, & \text{if }\gamma =n-(n-2)\beta , \\[3pt]
\frac{1}{(1+|x| )^{n-2}}, & \text{if }\gamma>n-(n-2)\beta .
\end{cases}
\]


\textbf{Case 3:} $\lambda =n-(n-2)\sigma $.
We have
\[
b(x)u^{p}(x)\approx \frac{L_1(1+|x| )}{(1+|x| )^{\mu +(n-2)p}}
\Big( \int_1^{|x| +2}
\frac{L(s)}{s}ds\Big) ^{p/(1-\sigma)}
:=\frac{L_{2}(1+|x| )}{(1+|x| )^{\mu +(n-2)p}}.
\]
By Lemma \ref{lem2} and Remark \ref{rmk4},  obviously we have that
$L_{2}\in \mathcal{K}$. Now suppose that $\mu +(n-2)p\geq 2$
and $\int_1^{\infty }t^{1-\mu -(n-2)p}L_{2}(t)dt<\infty $.
Then applying Theorem \ref{thm1}, we conclude that \eqref{4.3} has a
unique classical solution $(u,v) $ such that
$u(x)\approx \theta _{\lambda }(x)$
and
\[
v(x)\approx \begin{cases}
\big( \int_{|x| +1}^{\infty }\frac{L_{2}(s)}{s}
ds\big) ^{1/(1-\beta )}, & \text{if }\mu +(n-2)p=2, \\[3pt]
\frac{( L_{2}(1+|x| )) ^{1/(1-\beta )}
}{(1+|x|) ^{\frac{\mu +(n-2)p}{1-\beta }}},
& \text{if }2<\mu +(n-2)p<n-(n-2)\beta , \\[3pt]
\frac{1}{(1+|x|) ^{n-2}}\big(
\int_1^{|x| +2}\frac{L_{2}(s)}{s}ds\big) ^{1/(1-\beta)},
& \text{if }\mu +(n-2)p=n-(n-2)\beta , \\[3pt]
\frac{1}{(1+|x|) ^{n-2}}, & \text{if }\mu +(n-2)p>n-(n-2)\beta .
\end{cases}
\]

\textbf{Case 4:}  $\lambda >n-(n-2)\sigma $.
We have
\[
b(x)u^{p}(x)\approx \frac{L_1(1+|x|) }{
(1+|x|) ^{n-2+\mu }}.
\]
Suppose that $n-2+\mu \geq 2$ and
$\int_1^{\infty }t^{1-(n-2+\mu )}L_1(t)dt<\infty $.
Then applying Theorem \ref{thm1}, we conclude that \eqref{4.3}
has a unique classical solution $(u,v)$ such that
$u(x)\approx \theta _{\lambda }( x)$
and
\[
v(x)\approx \begin{cases}
\big( \int_{|x| +1}^{\infty }\frac{L_1(s) }{s}ds\big) ^{1/(1-\beta )},
 & \text{if }n-2+\mu =2, \\[3pt]
\frac{( L_1(1+|x| )) ^{1/(1-\beta )}
}{(1+|x|) ^{(\mu +n-4)/(1-\beta)}},
& \text{if }2<n-2+\mu <n-(n-2)\beta , \\[3pt]
\frac{1}{(1+|x|) ^{n-2}}
\big(\int_1^{|x| +2}\frac{L_1(s)}{s}ds\big)
^{1/(1-\beta)}, & \text{if }n-2+\mu =n-(n-2)\beta , \\
\frac{1}{(1+|x|) ^{n-2}}, & \text{if } n-2+\mu >n-(n-2)\beta .
\end{cases}
\]

\subsection*{Acknowledgments}
We want thank the anonymous referee for a careful reading of the
original manuscript.

\begin{thebibliography}{00}

\bibitem{SON} S. Ben Othman, H. M\^{a}agli, S. Masmoudi,
M. Zribi;
\emph{Exact asymptotic behavior near the boundary to the solution
for singular nonlinear Dirichlet problems},
Nonlinear Anal. 71 (2009) 4137-4150.

\bibitem{br} H. Brezis, S. Kamin;
\emph{Sublinear elliptic equations in $\mathbb{R}^n$},
Manuscripta Math. 74 (1992) 87-106.

\bibitem{cr} M. G. Crandall, P. H. Rabinowitz, L. Tartar;
\emph{On a Dirichlet problem with a singular nonlinearity},
Comm. Partial Differential Equations 2 (1977) 193-222.

\bibitem{ed} A. L. Edelson;
\emph{Entire solutions of singular elliptic equations},
J. Math. Anal. 139 (1989) 523-532.

\bibitem{rad} M. Ghergu, V. D. Radulescu;
\emph{Ground state solutions for the singular Lane-Emden-Fowler
equation with sublinear convection term}, J. Math.
Anal. Appl. 333 (2007) 265-273.

\bibitem{sy} S. Gontara, H. M\^{a}agli, S. Masmoudi, S. Turki;
\emph{Asymptotic behavior of positive solutions of a singular
nonlinear Dirichlet problem}, J.
Math. Anal. Appl. 369 (2010) 719-729.

\bibitem{ls} A. V. Lair, A. W. Shaker;
\emph{Classical and weak solutions of a
singular semilinear elliptic problem}, J. Math. Anal. Appl. 211 (1997)
371-385.

\bibitem{ma} H. M\^{a}agli;
\emph{Asymptotic behavior of positive solution of a
semilinear Dirichlet problem}, Nonlinear Anal. 74 (2011) 2941-2947.

\bibitem{mz} H. M\^{a}agli, M. Zribi;
\emph{Existence and estimates of solutions
for singular nonlinear elliptic problems}, J. Math. Anal. Appl. 263 (2001)
522-542.

\bibitem{MO} A. Mohammed;
\emph{Ground state solutions for singular semi-linear
elliptic equations}, Nonlinear Anal. 71 (2009) 1276-1280.

\bibitem{SC} S. C. Port, C. J. Stone;
\emph{Brownian motion and classical potential
theory}, Academic Press, San Diego, 1978.

\bibitem{san} C. A. Santos;
\emph{On ground state solutions for singular and
semilinear problems including super linear terms at infinity},
Nonlinear Anal. 71 (2009$)$ 6038--6043.

\bibitem{SEN} R. Seneta;
\emph{Regular varying functions, Lecture Notes in Math.}
Vol. 508. Springer-Verglag, Berlin 1976.

\bibitem{20} Z. Zhang;
\emph{The asymptotic behaviour of the unique solution for
the singular Lane-Emden-Fowler equation},
J. Math. Anal. Appl. 312 (2005) 33-43.

\end{thebibliography}

\end{document}