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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 81, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/81\hfil Liouville-type theorems for stable solutions]
{Liouville-type theorems for stable solutions of singular
quasilinear elliptic equations in $\mathbb{R}^N$}

\author[C. Chen, H. Song, H. Yang \hfil EJDE-2018/81\hfilneg]
{Caisheng Chen, Hongxue Song, Hongwei Yang}

\address{Caisheng  Chen (corresponding author) \newline
College of Science, Hohai University,
Nanjing 210098,  China}
\email{cshengchen@hhu.edu.cn}

\address{Hongxue Song \newline
College of Science, Hohai University,
Nanjing 210098,  China.\newline
College of Science,
Nanjing University of Posts and Telecommunications,
Nanjing 210023, China}
\email{songhx@njupt.edu.cn}

\address{Hongwei Yang \newline
College of Mathematics and System Science,
Shandong University of Science and Technology,
Qingdao 266590, China}
\email{hwyang1979@163.com}

\dedicatory{Communicated by Vicentiu D. Radulescu}

\thanks{Submitted June 26, 2017. Published March 22, 2018.}
\subjclass[2010]{35J60, 35B53, 35B33, 35B45}
\keywords{Singular quasilinear elliptic equation; stable solutions; 
\hfill\break\indent critical exponents;
Liouville type theorems}

\begin{abstract}
 We prove a Liouville-type theorem for stable solution of the singular
 quasilinear elliptic equations
 \begin{gather*}
 -\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)|u|^{q-1}u, \quad
 \text{in } \mathbb{R}^N, \\
 -\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)e^u, \quad
 \text{in } \mathbb{R}^N
 \end{gather*}
 where  $2\le p<N$, $-\infty<a<(N-p)/p$ and the function  $f(x)$ is continuous
 and nonnegative in $\mathbb{R}^N\setminus\{0\}$ such that
 $f(x)\ge c_0|x|^{b}$ as $|x|\ge R_0$, with $b>-p(1+a)$ and $c_0>0$.
 The results hold for $1\le p-1<q=q_c(p,N,a,b)$ in the first equation,
 and for $2\le N<q_0(p,a,b)$ in the second equation.
 Here $q_0$ and $q_c$ are exponents, which are always larger than the
 classical critical ones and depend on the parameters $a,b$.
\end{abstract}

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

\section{Introduction and main results}

Recently, Ghergu and R\u{a}dulescu \cite{S19} studied  the
singular elliptic problem
 \begin{equation}\label{1.1}
 -\operatorname{div}(|x|^{-2a}\nabla u)=K(x)|x|^{-bq}|u|^{q-2}u+\lambda g(x),
 \quad  x\in  \mathbb{R}^N,
 \end{equation}
where $-\infty<a<(N-2)/2$, $a\le b<a+1$, $q=2N/(N-2(1+a-b))$ and 
$N\ge 2$. Under some natural assumptions on the positive potential $K(x)$,
the authors established the existence of some $\lambda_0>0$ such
that the problem \eqref{1.1} has at least two distinct solutions
provided that $\lambda\in (0,\lambda_0)$. For $\lambda=0$, there
exists an interesting question: does  \eqref{1.1} admit a
nontrivial solution?

D'Ambrosio and Mitidieri \cite{S7} considered the existence and nonexistence
of nontrivial weak solution to the following quasilinear elliptic
equation with singular weights and  critical exponent
\begin{equation}\label{1.2}
\operatorname{div}(A(x,u,\nabla u)) + V(x)|u|^{p-2}u = a(x)|u|^{q-1}u,
\quad x\in\mathbb{R}^N
 \end{equation}
Here $A$ contains the $p-$Laplacian operator
$A(x,t,\xi)=|\xi|^{p-2}\xi$ and  the mean curvature operator
$A(x,t,\xi)=\xi/\sqrt{1+|\xi|^2}$ for $\xi\in \mathbb{R}^N$,
 $p>1$, $V\ge 0$ is a singular potential function,
$a(x):\mathbb{R}^N\to \mathbb{R}$ is a nonnegative measurable
function and $q>p-1$. Similar consideration can be found
\cite{S1,S2,S16,S17,S18,S20}  and the references therein.

 In this article, motivated by Chen \cite{S5}, Dancer et al.\
 \cite{S8} and the references mentioned above, we study the 
nonexistence of stable
solutions to the singular quasilinear elliptic  equation
\begin{equation}\label{1.3}
-\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=g(x,u), \quad 
\text{in } \mathbb{R}^N.
\end{equation}
In particular,  we are interested in  the
Liouville-type theorems for stable solutions of the singular
quasilinear elliptic  equations
 \begin{gather}\label{1.4}
-\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)|u|^{q-1}u, \quad
\text{in } \mathbb{R}^N, \\
\label{1.5}
-\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)e^u, \quad 
\text{in } \mathbb{R}^N,
\end{gather}
where $2\le p<N$, $-\infty<a<(N-p)/p$,  $f(x)\in C(\mathbb{R}^N\setminus\{0\})$ 
is nonnegative. The exact assumption on
$f(x)$ will be given in (H1) below.

For $a \neq 0 $ and $p\neq 2$,  to our knowledge, 
there is very little information on the  nonexistence of stable solutions
for  problems \eqref{1.4} and \eqref{1.5}.

  In this article, we are concerned about stable solutions of
  \eqref{1.4} and \eqref{1.5}
in the following sense.

\begin{definition}[\cite{S7,S27}] \label{def1.1} \rm
 Let  $g(x,\cdot):\mathbb{R}\to\mathbb{R}$ be a $C^1$  function for almost every 
$x\in \mathbb{R}^N$.   We say that $u$ is a weak solution
 of \eqref{1.3} if $u\in C^{1,\omega}_{\rm loc}(\mathbb{R}^N)(0<\omega<1)$
satisfies $g(x,u)\in L^1_{\rm loc}(\mathbb{R}^N)$ and
\begin{equation}\label{1.6}
\int_{\mathbb{R}^N}|x|^{-ap}|\nabla u|^{p-2}\nabla u\nabla\zeta\,dx
=\int_{\mathbb{R}^N}g(x,u)\zeta \,dx,\quad \forall \zeta\in C_0^1(\mathbb{R}^N).
\end{equation}
Let $u$ be a weak solution of \eqref{1.3}. We say that $u$ is stable
if $g_u(x,u)\in L^1_{\rm loc}(\mathbb{R}^N)$ and
\begin{equation}\label{1.7}
\begin{aligned}
Q_u(\zeta)&:=\int_{\mathbb{R}^N}  |x|^{-ap}\Big(|\nabla
u|^{p-2}|\nabla\zeta|^2+(p-2)|\nabla u|^{p-4}(\nabla
u\cdot\nabla\zeta)^2\Big)dx \\
&\quad -\int_{\mathbb{R}^N}  g_u(x,u)\zeta^2dx\ge 0,
\end{aligned}
\end{equation}
for every $\zeta\in C_0^1(\mathbb{R}^N)$.

The Morse index of a solution $u$, $i(u)$ is defined as the maximal
dimension of all subspace $X$ of $C_0^1(\mathbb{R}^N)$ such that
$Q_u(\zeta)<0$ for any $\zeta\in X\setminus\{0\}$. Clearly, $u$ is
stable if and only if $i(u)=0$.
\end{definition}

We note  that the $C^{1,\omega}$ regularity assumption is natural to
the  solution of \eqref{1.3}  due to the results in
\cite{S1,S8,S10,S29}.

\begin{remark}  \label{rmk1.1} \rm
If  $u$ is  a stable weak solution of \eqref{1.5}, then  from  \eqref{1.7} 
it follows  that
\begin{equation}\label{1.8}
\int_{\mathbb{R}^N}f(x)e^u\zeta^2dx\le (p-1)\int_{\mathbb{R}^N}|x|^{-ap}|\nabla
u|^{p-2}|\nabla\zeta|^2dx,\quad \forall \zeta\in C_0^1(\mathbb{R}^N).
\end{equation}
Similarly, if $u$ is a stable  nonnegative solution of \eqref{1.4},
we have from \eqref{1.7} that
\begin{equation}\label{1.9}
q\int_{\mathbb{R}^N}f(x)u^{q-1}\zeta^2dx\le (p-1)\int_{\mathbb{R}^N}|x|^{-ap}|\nabla
u|^{p-2}|\nabla\zeta|^2dx,\quad \forall \zeta\in C_0^1(\mathbb{R}^N).
\end{equation}
\end{remark}

 We recall that Liouville-type theorem is the
nonexistence of nontrivial solution in the entire space $\mathbb{R}^N$. The
classical Liouville theorem stated that a bounded harmonic (or
holomorphic) function defined in entire space $\mathbb{R}^N$ must be
constant. This theorem, known as Liouville Theorem, was first
announced in 1844 by Liouville \cite{S24} for the special case of a
doubly-periodic function. Later in the same year, Cauchy \cite{S4}
published the first proof of the above stated theorem. In 1981,
Gidas and Spruck established in pioneering article \cite{S21} the
optimal Liouville type result for nonnegative solutions to the
singular equation \eqref{1.4} with $p=2, a=0$ and $f(x)=|x|^b$:
\begin{equation}\label{1.10}
-\Delta u=|x|^{b}|u|^{q-1}u,\quad x\in\mathbb{R}^N.
\end{equation}
They proved that  \eqref{1.10} with $b=0$ has no positive solution if and only if
$1<q<q_s=\frac{N+2}{N-2}$ if $N>2$ and $q_s=\infty$  if $N=2$.

The case $b\not=0$ is less completely understood. Let us first
recall that if $b\le -2$, then \eqref{1.10} has no positive solution
in any domain $\Omega$ containing the origin, see \cite{S3,S21}. We
therefore restrict ourselves to the case $b>-2$ in the rest of this
article. Let us introduce the Hardy-Sobolev exponent
\begin{equation}\label{1.11}
q_s(b)=\frac{N+2+2b}{N-2}(=\infty),\quad\text{if } N=2.
\end{equation}
In the class of radial
solutions, the Liouville property was completely solved
\cite{S3,S26}.

\begin{proposition} \label{prop1.2}
 Let $N\ge 2, b>-2$ and $q>1$.
\begin{itemize}
\item[(i)] If $q<q_s(b)$, then \eqref{1.10} has no positive radial solution
in $\mathbb{R}^N$.

\item[(ii)] If $q\ge q_s(b)$, then \eqref{1.10} possesses a bounded,
positive radial solution in $\mathbb{R}^N$.
\end{itemize}
\end{proposition}

So far, for the radial solutions, the results have been clean and
neat. On the other hand,  using Farina's approach in \cite{S13},
Fazly \cite{S15} established Liouville type theorem of the weighted
Lane-Emden equation
\begin{equation}\label{1.12}
-\Delta u=(1+|x|^2)^{b/2}|u|^{q-1}u,\quad \text{in }\mathbb{R}^N.
\end{equation}

\begin{proposition}[{\cite[ Theorems 2.3]{S15}}]  \label{prop1.3} 
Let  $u$ be a nonnegative entire semi-stable solution of \eqref{1.12} with $b>-2,
q\ge 2$. Then $u$ is the trivial solution if the space dimension $N$
satisfies
\begin{equation}\label{1.13}
2\le N<2+\frac{2(2+b)}{q-1}\Big(q+\sqrt{q^2-q}\Big).
\end{equation}
\end{proposition}

\begin{remark}  \label{rmk1.4} \rm
Obviously,  If $b>-2$ and $N>10+4b$, then \eqref{1.13} implies
that
\begin{equation}\label{1.14}
\begin{aligned}
1&<q<q_c\\
&:=\frac{(N-2)(N-6-2b)-2(2+b)^2+2(2+b)\sqrt{(2+b)(2N-2+b)}}{(N-2)(N-10-4b)}.
\end{aligned}
\end{equation}
\end{remark}

Similar works can be founded in \cite{ S9,S11,S20,S23,S28} and the
references therein.

To our knowledge, there are only few works on  exponential case
\eqref{1.5} as compared with \eqref{1.10} and \eqref{1.12}. Farina
in \cite{S14} proved  that $\Delta u+e^u=0$ has no stable classical
solution in $\mathbb{R}^N$ for $2\le N\le 9$. Dancer and farina in \cite{S8}
proved  that \eqref{1.5} with $p=2$ and $f(x)=1$ admits classical entire
 solutions which are stable outside a compact set if and only if 
$N\ge  10$. Recently, Wang and Ye in \cite{S28} proved

 \begin{theorem} \label{thm1.5}
Let $p=2,a=0$ and $f(x)=|x|^{b}$ with $b>-2$. For $2\le N<10+4b$,
there is no weak stable solution of \eqref{1.5}.
\end{theorem}

 In this paper,  the first aim is to show the
nonexistence of stable solutions to \eqref{1.5} with the weighted
functions $f(x)$ and $2\le p<N$. Since $p>2$, the test functions in
the above references does not work. For the estimation of solution,
we need to choose some special test functions to investigate our
problem.

Throughout this paper, we make the following assumption on $f(x)$.
\begin{itemize}
 \item[(H1)] $f(x)\in C(\mathbb{R}^N\setminus\{0\})$  is  nonnegative in 
$\mathbb{R}^N$. In addition, there
exist $b>-p(1+a), c_0>0$ and $R_0>0$  such that 
$ f(x)\ge c_0|x|^{b}$ for all $|x|\ge R_0$.
\end{itemize}
Denote
\[
q_0(p,a,b)=\frac{p(p+3)(1+a)+4b}{p-1}.
\]
Our main results in this paper are as follows. 

\begin{theorem}  \label{thm1.6}
 Suppose that the function $f(x)$ satisfies {\rm (H1)} and  $2\le p<N<q_0(p,a,b).$
Then there is no weak stable solution of \eqref{1.5}.
\end{theorem}

\subsection*{Open problem}  When $N>q_0(p,a,b)$ or $1<p<2$, does equation
\eqref{1.5} admit a stable  solution?

\begin{remark} \label{rmk1.7} \rm
If $p=2$, $a=b=0$, then $q_0(2,0,0)=10$. The result in Theorem \ref{thm1.6}
coincides with that in \cite{S14}. If $p=2$, $a=0$ and $b>-2$,
$q_0(2,0,b)=10+4b$. It is the critical exponent $q_c=10+4b$ in
\cite{S28}.
\end{remark}

\begin{theorem} \label{thm1.8}  
 Suppose that the function $f(x)$ satisfies {\rm (H1)} and  $p\ge 2, b>-p(1+a)$.
 Let $u\in C_{\rm loc}^{1\delta}(\mathbb{R}^N)$ be a stable  solution of problem
\eqref{1.4}.
 Assume that
\begin{equation}\label{1.15}
 \begin{cases} 
p-1<q<\infty,\quad & \text{if } N\le q_0(p,a,b), \\
p-1<q<q_c=q_c(p,a,b), &\text{if } N>q_0(p,a,b)
\end{cases}
\end{equation}
with the critical exponent
\begin{equation}\label{1.16}
\begin{split}
&q_c(p,N,a,b) \\
&=\Big((p - 1)[N^2(p - 1) - p(1+a)(N(p + 2) - p(1 + a)) \\
&\quad + b(N(p - 4) - p^2(1 + a)) -2b^2]\Big) \\
&\quad\div \Big((N-p(1+a))[N(p-1)-p(p+3)(1+a)-4b]\Big)\\
&\quad +\frac{2(p(1+a)+b)\sqrt{(p-1)(p(1+a)+b)[p(N-1-a)+b(p-1)]}}{(N-p(1+a))
 [N(p-1)-p(p+3)(1+a)-4b]}.
\end{split}
\end{equation}
Then $u\equiv 0$ in $\mathbb{R}^N$.
\end{theorem}

\begin{remark} \label{rmk1.9} \rm
If $a=0$, then
\begin{equation}\label{1.17}
\begin{split}
&q_c(p,N,0,b) \\
&=\frac{(p-1)[N^2(p-1)-p(N(p+2)-p)+b(N(p-4)-p^2)-2b^2]}{(N-p)[(N(p-1)-p(p+3)-4b]}
\\
&\quad +\frac{2(p+b)\sqrt{(p+b)(p-1)[p(N-1)+b(p-1)]}}{(N-p)[(N(p-1)-p(p+3)-4b]}.
\end{split}
\end{equation}
It is the critical exponent $q_c$ in \cite{S5}. Furthermore, if
$a=b=0$, then
 \begin{equation}\label{1.18}
\begin{aligned}
&q_c(p,N,0,0) \\
&=\frac{(p-1)[N^2(p-1)-p(N(p+2)-p)]
+2p^2\sqrt{(p-1)(N-1)}}{(N-p)[(N(p-1)-p(p+3)]}.
\end{aligned}
\end{equation}
It equals  the critical exponent $p_c$ in \cite{S6}. Also, we
observe that the critical exponent $q_c(p,N,0,0)$ is always greater
than the classic critical exponent $\frac{N(p-1)+p}{N-p}$. If
$a=b=0$ and $p=2$, we find
\begin{equation}\label{1.19}
q_c(2,N,0,0)=\frac{N^2-8N+4+8\sqrt{N-1}}{(N-2)(N-10)}.
\end{equation}
It is  the critical exponent $p_c$ in \cite{S13} and the exponent
$q_c(p,N)$ in \cite{S22} and $p(N,\alpha)$ in \cite{S12}, and
coincides with that
 in \cite{S25}.
\end{remark}

\begin{remark}  \label{rmk1.10} \rm
Clearly, problems \eqref{1.4} and \eqref{1.5} are an extension
of problems in \cite{S14,S22,S25,S28} respectively. Our conclusions
in Theorems \ref{thm1.6} and \ref{thm1.8} extend results in the above references.
\end{remark}

\begin{remark}  \label{rmk1.11} \rm
 For \eqref{1.1}, we let $\lambda=0$ and $N(1+\frac{8b}{N-2(1+a-b)})<10(1+a)$. Then,
 an application of Theorem \ref{thm1.8}
shows that there are no stable solutions.
\end{remark}

 The rest of the paper is devoted to the proof of Theorems \ref{thm1.6} and 
\ref{thm1.8}.  In the following, we denote by $C_j$ $(j=1,2,\dots)$
positive constants, which may vary from line to line.

\section{Proof of Theorem \ref{thm1.6}}

To prove the nonexistence of solutions to \eqref{1.5}, we
use the test function method, which has been used in \cite{S6,S14}
and references therein. Since $2\le p<N$, some modification in
choosing functions is necessary.  The proof is based on argument by
contradiction which involves a priori estimate for a solution of
\eqref{1.5} by carefully choosing the special test function and
scaling argument.

For any nonnegative function $\varphi\in C_0^1(\mathbb{R}^N)$ and
$\alpha>0$, we denote $\zeta=e^{p\alpha u}\varphi^p$. Then, it
follows from \eqref{1.5} and \eqref{1.6} that
\begin{equation}\label{2.1}
\begin{aligned}
&\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha u}|\nabla u|^p\varphi^pdx \\
&=\frac{1}{p\alpha}\int_{\mathbb{R}^N}  f(x)e^{(p\alpha+1)u}\varphi^pdx
-\frac{1}{\alpha}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha u}
 \varphi^{p-1}|\nabla u|^{p-2}\nabla u\cdot\nabla \varphi dx.
\end{aligned}
\end{equation}

By Young inequality with any $\varepsilon>0$, one sees that
\begin{equation}\label{2.2}
\begin{split}
&\frac{1}{\alpha}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha
u}\varphi^{p-1}|\nabla u|^{p-1}|\nabla \varphi|dx  \\
&\le \varepsilon\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha u}|\nabla
u|^p\varphi^pdx+C_{\varepsilon}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha
u}|\nabla\varphi|^pdx
\end{split}
\end{equation}
It follows from \eqref{2.1} and \eqref{2.2} that
\begin{equation}\label{2.3}
\begin{split}
&(1-\varepsilon)\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha
u}\varphi^{p-1}|\nabla u|^{p-1}|\nabla \varphi|dx \\
&\le \frac{1}{p\alpha}\int_{\mathbb{R}^N}f(x)e^{(p\alpha+1)u}\varphi^pdx
+C_{\varepsilon}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha u}|\nabla\varphi|^pdx
\end{split}
\end{equation}

On the other hand, we take $\zeta=(e^{p\alpha u}\varphi^p)^{1/2}$ in
\eqref{1.8} and obtain
\begin{equation}\label{2.4}
\begin{split}
&\int_{\mathbb{R}^N} f(x)e^{(p\alpha+1)u}\varphi^pdx \\
&\le (p-1)\int_{\mathbb{R}^N} |x|^{-ap}|\nabla u|^{p-2}|\nabla(e^{p\alpha
u/2}\varphi^{p/2})|^2dx \\
&=\frac{p^2(p-1)}{4}\int_{\mathbb{R}^N} |x|^{-ap}e^{p\alpha
u}\Big(\alpha^2|\nabla u|^p\varphi^p+2\alpha\varphi^{p-1}|\nabla
u|^{p-2}\nabla u\nabla\varphi  \\
&\quad +|\nabla u|^{p-2}|\nabla\varphi|^2\varphi^{p-2}\Big)dx
\end{split}
\end{equation}
Similarly, by Young inequality, we have
\begin{equation}\label{2.5}
\begin{split}
&\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha u}\varphi^{p-2}|\nabla
u|^{p-2}|\nabla \varphi|^2dx \\
&\le
\varepsilon\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^pe^{p\alpha
u}\varphi^pdx+C_{\varepsilon}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha
u}|\nabla\varphi|^pdx
\end{split}
\end{equation}
Then an application of \eqref{2.3}-\eqref{2.5} gives
\begin{equation}\label{2.6}
\begin{split}
&\int_{\mathbb{R}^N}  f(x)e^{(p\alpha+1)u}\varphi^{p}dx \\
&\le \frac{p^2(p-1)(\alpha^2+\varepsilon+2\alpha\varepsilon)}{4}
 \int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^pe^{p\alpha
u}\varphi^pdx \\
&\quad +C_{\varepsilon}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha
u}|\nabla\varphi|^pdx
\\
&\le\frac{p^2(p-1)(\alpha^2+\varepsilon+2\alpha\varepsilon)}
 {4p\alpha(1-\varepsilon)}\int_{\mathbb{R}^N}f(x)e^{(p\alpha+1)u}\varphi^{p}dx\\
&\quad +C_{\varepsilon}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha
u}|\nabla\varphi|^pdx.
\end{split}
\end{equation}
Then, one sees that
\begin{equation}\label{2.7}
\lambda_1\int_{\mathbb{R}^N}f(x)e^{(p\alpha+1)u}\varphi^{p}dx\le
C_{\varepsilon}\int_{\mathbb{R}^N}  |x|^{-ap}e^{p\alpha
u}|\nabla\varphi|^pdx,
\end{equation}
where
\begin{equation}\label{2.8}
\lambda_1=\lambda_0-\frac{\lambda_2\varepsilon}{1-\varepsilon},\quad
\lambda_0=1-\frac{p(p-1)\alpha}{4},\quad
\lambda_2=\frac{p(p-1)(1+\alpha)^2}{4\alpha}.
\end{equation}
in which we choose $0<\alpha< \frac{4}{p(p-1)}$ such that
$\lambda_0>0$. Furthermore, let $\varepsilon>0$ be so small that
$\lambda_1>0$.

By H\"{o}lder inequality,  from \eqref{2.7} we obtain
\begin{equation}\label{2.9}
\begin{aligned}
\int_{\mathbb{R}^N}  fe^{(p\alpha+1)u}\varphi^{p}dx 
&\le C_{\varepsilon}\Big(\int_{\mathbb{R}^N}  
 fe^{(p\alpha+1)u}\varphi^pdx\Big)^{\frac{p\alpha}{p\alpha+1}} \\
&\quad\times  \Big(\int_{\mathbb{R}^N}  f^{-p\alpha}|x|^{-pa(p\alpha+1)}
 \varphi^{-p^2\alpha}|\nabla
\varphi|^{p(p\alpha+1)}dx\Big)^{\frac{1}{p\alpha+1}},
\end{aligned}
\end{equation}
where $f=f(x)$. This implies 
\begin{equation}\label{2.10}
\int_{\mathbb{R}^N}  f(x)e^{(p\alpha+1)u}\varphi^{p}dx\le C_{\varepsilon}
\int_{\mathbb{R}^N}  f^{-p\alpha}|x|^{-pa(p\alpha+1)}\varphi^{-p^2\alpha}|\nabla
\varphi|^{p(p\alpha+1)}dx
\end{equation}

We now choose $\varphi_0(s)\in C_0^1[0,\infty)$ defined by
\begin{equation}\label{2.11}
 \varphi_0(s)=\begin{cases}
1,& 0\le s\le 1, \\
2(2-s)^k-(2-s)^{2k},& 1<s<2, \\
0, & s>2,
\end{cases}
 \end{equation}
where $k=p\alpha+1>1$. It is not difficult to verify that 
$0\le \varphi_0(s)\le 1$ and 
$|\varphi'_0(s)|\le \beta_0\varphi_0^{1-1/k}(s)$ with $\beta_0=2^{1/k}k$.

We let $\varphi=\varphi(x)=\varphi_0(\frac{|x|}{R})$ with $R>R_0$,
where $R_0$ is given in (H1). Then, setting $x=R\xi$, we get
\begin{equation}\label{2.12}
\begin{split}
\int_{\mathbb{R}^N}  f^{-p\alpha}\varphi^{-p^2\alpha}|\nabla
\varphi|^{p(p\alpha+1)}dx
&\le CR^{\theta}\int\limits_{1\le|\xi|\le 2}
 \Big(\frac{|\varphi_0'(|\xi|)}{\varphi_0(|\xi|)^{1-\frac{1}{p\alpha+1}}}
 \Big)^{p(p\alpha+1)}d\xi \\
&\le CR^{\theta}\beta_0^{p(p\alpha+1)}
\end{split}
\end{equation}
where $C$ is a positive constant independent of $R$ and
$\theta=N-p(1+a)-p\alpha(b+p(1+a))$. Noticing that
 $0<p(p-1)\alpha<4$ and $N<q_0(p,a,b)=\frac{p(p+3)(1+a)+4b}{p-1}$, 
we can so choose that $\alpha$ such that $N<p(1+a)+p\alpha(b+p(1+a))$.
  Then, letting $R\to +\infty$, we obtain from
\eqref{2.12} that
\begin{equation}\label{2.13}
\int_{\mathbb{R}^N}  f(x)e^{(p\alpha+1)u}dx=0
\end{equation}
This is impossible. This completes the proof.

\section{Proof of Theorem \ref{thm1.8}}

 We first establish some estimation for solutions of \eqref{1.4}.

\begin{lemma} \label{lem3.1}
Let $u\in C_{\rm loc}^{1,\omega}(\mathbb{R}^N)$ be a stable weak solution of
\eqref{1.4} with $q>p-1\ge 1$. Then for every $k\in (1, \; k_0(q))$,
where
\begin{equation}\label{3.1}
k_0(t)=\frac{2t-p+1+2\sqrt{t(t-p+1)}}{p-1},\quad t>p-1,
\end{equation}
and for any integer $n$ with $ n\ge
\max\{2,\frac{q+k}{q-p+1}\} $, there exists a constant
$C=C(q,p,n,k)$ such that
\begin{equation}\label{3.2}
\begin{aligned}
&\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^{pn}dx
+\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^p|u|^{k-1}\varphi^{pn}dx \\
&\le C\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
\varphi|^{\frac{p(q+k)}{q-p+1}}|f(x)|^{-\frac{k+p-1}{q-p+1}}dx,
\end{aligned}
\end{equation}
where  $\varphi\in C_0^1(\mathbb{R}^N)$ is  the  nonnegative cut-off
function, in which $\varphi(x)=\varphi_0(\frac{|x|}{R})$ for any
$R>0$ with  $\varphi_0(s)\in C_0^1(\mathbb{R}^+), 0\le \varphi_0(s)\le 1$
and
\begin{equation}\label{3.3}
 \varphi_0(s)= \begin{cases} 
1, & 0\le s\le 1,\\
0, & s\ge 2.
\end{cases}
\end{equation}
\end{lemma}

\begin{proof}  
By the definition of $\varphi(x)$, we know that there exists $C>0$ such that
 $|\nabla \varphi(x)|\le CR^{-1}$ in $x\in
\overline{B}_{2R}\setminus \overline{B}_{R}$ and 
$|\nabla \varphi(x)|=0$ if $x\in B_R\cup B_{2R}^c$, where
$B_r=\{x\in\mathbb{R}^N:|x|<r\}$.

Let $u\in C^{1,\omega}_{\rm loc}(\mathbb{R}^N)$ be a stable solution of
\eqref{1.4} and $k>1$. Multiplying \eqref{1.4} by
$|u|^{k-1}u\varphi^p$ and integrating by parts, we find
\begin{equation}\label{3.4}
\begin{aligned}
&k\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^p|u|^{k-1}\varphi^pdx \\
&\le p\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^{p-1}|\nabla
\varphi||u|^k\varphi^{p-1}dx+\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^pdx.
\end{aligned}
\end{equation}
 Then applying  Young's inequality with parameter $\epsilon\in (0,1)$, we have
\begin{equation}\label{3.5}
\begin{aligned}
&p\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^{p-1}|\nabla
\varphi||u|^k\varphi^{p-1}dx \\
&\le \epsilon\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
u|^p|u|^{k-1}\varphi^pdx+C_1\int_{\mathbb{R}^N}  
|x|^{-ap}|\nabla\varphi|^p|u|^{k+p-1}dx.
\end{aligned}
\end{equation}
 Then from \eqref{3.4} and \eqref{3.5} it follows  that
\begin{equation}\label{3.6}
\begin{aligned}
&(k-\epsilon)\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^p|u|^{k-1}\varphi^pdx \\
&\le \int_{\mathbb{R}^N}  f(x)|u|^{q+k}\varphi^pdx
 +C_1\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla\varphi|^p|u|^{k+p-1}dx.
\end{aligned}
\end{equation}
On the other hand, taking
$\zeta=|u|^{\frac{k-1}{2}}u\varphi^{\frac{p}{2}}$ in \eqref{1.9},
one sees that
\begin{equation}\label{3.7}
\begin{split}
&\frac{q}{p-1}\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^pdx \\
&\le \frac{1}{4}(1+k)^2\int_{\mathbb{R}^N} |x|^{-ap}|\nabla
u|^p|u|^{k-1}\varphi^pdx\\
&\quad +\frac{p^2}{4}\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^{p-2}|\nabla
\varphi|^2|u|^{k+1}\varphi^{p-2}dx \\
&\quad +\frac{1}{2}p(1+k)\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
u|^{p-1}|\nabla \varphi||u|^k\varphi^{p-1}dx.
\end{split}
\end{equation}
By Young's inequality with  $\epsilon>0$, we obtain
\begin{gather}\label{3.8}
\begin{aligned}
&\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^{p-2}|\nabla
\varphi|^2|u|^{k+1}\varphi^{p-2}dx \\
&\le \epsilon\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
u|^p|u|^{k-1}\varphi^pdx+C_2\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
\varphi|^{p}|u|^{k+p-1}dx,
\end{aligned} \\
\label{3.9}
\begin{aligned}
&\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^{p-1}|\nabla \varphi||u|^k\varphi^{p-1}dx \\
&\le \epsilon\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
u|^p|u|^{k-1}\varphi^pdx+C_3\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
\varphi|^{p}|u|^{k+p-1}dx.
\end{aligned}
\end{gather}
Then it follows from \eqref{3.7}-\eqref{3.9} that
\begin{equation}\label{3.10}
\begin{aligned}
\frac{q}{p - 1} \int_{\mathbb{R}^N}  f(x)|u|^{q+k}\varphi^pdx
&\le \beta_{\epsilon}\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
u|^p|u|^{k-1}\varphi^pdx \\
&\quad +C_4\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
\varphi|^{p}|u|^{k+p-1}dx
\end{aligned}
\end{equation}
with $\beta_{\epsilon}=\frac{1}{4}[(1+k)^2+p^2\epsilon+2p(1+k)\epsilon]>0$.
Furthermore, we obtain from \eqref{3.6} and \eqref{3.10} that
\begin{equation}\label{3.11}
\alpha_{\epsilon}\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^pdx\le
C_5\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla \varphi|^{p}|u|^{k+p-1}dx
\end{equation}
with some constant $C_5>0$ and
\begin{equation}\label{3.12}
\alpha_{\epsilon}=\frac{q}{p-1}-\frac{\beta_{\epsilon}}{k-\epsilon},\quad
\lim_{\epsilon\to 0^+}\alpha_{\epsilon}=\alpha_0
:=\frac{q}{p-1}-\frac{(1+k)^2}{4k}.
\end{equation}
The fact $\alpha_0>0$ implies that $k\in (1, k_0(q))$. Now, the
application of \eqref{3.6} and \eqref{3.11} yields
\begin{equation}\label{3.14}
\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla u|^{p}|u|^{k-1}\varphi^pdx
\le C_5\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla \varphi|^{p}|u|^{k+p-1}dx.
\end{equation}

We claim that the estimate \eqref{3.2} holds. Choose the
integer $n\ge \max\{2,\frac{q+k}{q-p+1}\} $. Then  one sees that
\begin{equation}\label{3.16}
\varphi^{\frac{p(n-1)(q+k)}{k+p-1}}(x)\le \varphi^{pn}(x),\quad x\in
\mathbb{R}^N.
\end{equation}

Then, replacing $\varphi$ by $\varphi^n$ in \eqref{3.11}, we find
\begin{equation}\label{3.17}
\begin{split}
&\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^{pn}dx \\
&\le \Big(\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^{p(n-1)\lambda}dx
 \Big)^{\frac{1}{\lambda}}
\Big(\int_{\mathbb{R}^N}f^{-\frac{\lambda'}{\lambda}}|x|^{-pa\lambda'}|\nabla
\varphi|^{p\lambda'}dx\Big)^{\frac{1}{\lambda'}} \\
&\le \Big(\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^{pn}dx\Big)^{\frac{1}{\lambda}}
\Big(\int_{\mathbb{R}^N}f^{-\frac{\lambda'}{\lambda}}|x|^{-pa\lambda'}|\nabla
\varphi|^{p\lambda'}dx\Big)^{\frac{1}{\lambda'}},
\end{split}
\end{equation}
where $\lambda=(q+k)/(k+p-1)>1, \lambda'=(q+k)/(q-p+1)>1$. So, it
derives by \eqref{3.17} that
\begin{equation}\label{3.18}
\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^{pn}dx
\le C\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
\varphi|^{\frac{p(q+k)}{q-p+1}}f^{-\frac{k+p-1}{q-p+1}}dx,
\end{equation}
Similarly, replacing  $\varphi$  by
$\varphi^n$ in \eqref{3.14},  from \eqref{3.17} and
\eqref{3.18} we obtain
\begin{align*}
\int_{\mathbb{R}^N}  |x|^{-pa}|\nabla u|^{p}|u|^{k-1}\varphi^{pn}dx
&\le C \int_{\mathbb{R}^N}  |x|^{-pa}\varphi^{p(n-1)}|u|^{k+p-1}|\nabla
\varphi|^pdx\\
&\le C\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
\varphi|^{\frac{p(q+k)}{q-p+1}}f^{-\frac{k+p-1}{q-p+1}}dx.
\end{align*}
So, we obtain  \eqref{3.2} and the proof of Lemma \ref{lem3.1} is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.8}]
 From estimate \eqref{3.2} and the
definition of the function $\varphi(x)$, it follows that
\begin{equation}\label{3.19}
\int_{\mathbb{R}^N}f(x)|u|^{q+k}\varphi^{pn}dx
+\int_{\mathbb{R}^N}  |x|^{-ap}|\nabla
u|^p|u|^{k-1}\varphi^{pn}dx\le CR^\tau,
\end{equation}
where assumption (H1)  has been used and
\begin{equation}\label{3.20}
\tau=N-p(1+a)-\frac{(p(1+a)+b)(k+p-1)}{t-p+1}.
\end{equation}

 Clearly, if $\tau<0$, the desired result follows by letting
 $R\to\infty$ in \eqref{3.19}. In the following, we consider the case in which
 $\tau<0$. Define the function
\begin{equation}\label{3.22}
g(t)=\frac{k_0(t)+p-1}{t-p+1},\quad t>p-1,
\end{equation}
where $k_0(t)$ is given in \eqref{3.1}.  Obviously,
\begin{equation}\label{3.23}
\lim_{t\to(p-1)^+}g(t)=+\infty,\quad
\lim_{t\to+\infty}g(t)=g_{\infty}:=\frac{4}{p-1}.
\end{equation}
Since
\begin{equation}\label{3.24}
g'(t)=\frac{-1}{(t-p+1)^2}\Big[1+\frac{t-p+1}{\sqrt{t(t-p+1)}}\Big]<0,
\quad \text{for } t>p-1,
\end{equation}
the function $g(t)$ is decreasing in $t>p-1$. So, we have
$g_{\infty}<g(t)<+\infty$ for $t>p-1$.

Therefore, if $N-p(1+a)\le (p(1+a)+b)g_{\infty}$, then
$N-p(1+a)<(p(1+a)+b)g(t)$ for any $t>p-1$. Hence if we fix  $k\in
[1, k_0(t))$ suitably near $k_0(t)$, we obtain
\begin{equation}\label{3.25}
N-p(1+a)< \frac{(p(1+a)+b)(k+p-1)}{t-p+1}.
\end{equation}
For this reason, the desired result follows by letting $R\to\infty$
in \eqref{3.19}.

Assume now $N-p(1+a)>(p(1+a)+b)g_{\infty}$.  Since $g$ is
decreasing, we get in this case a critical value $q_c(p,N,a,b)$ such
that $N-p(1+a)<(p(1+a)+b)g(q)$ for $p-1<q<q_c(p,N,a)$.  From this,
the desired result follows again by letting $R\to\infty$ in
\eqref{3.19}. Clearly, $q_c(p,N,a,b)$ may be deduced from the
equation $N-p(1+a)=(p(1+a)+b)g(q)$, which is  given the value in
\eqref{1.16}. Then we complete the proof.
\end{proof}


\subsection*{Acknowledgments}
The authors would like to express their
sincere gratitude to the reviewers for their valuable comments and
suggestions. This work is supported by the China Postdoctoral
Science Foundation (No. 2017M611664).

\begin{thebibliography}{99}

\bibitem{S1} R. B. Assunc\~ao,  P. C. Carri\~ao, O. H. Miyagaki;
\emph{Subcritical perturbations of a singular quasilinear
elliptic equation involving the critical Hardy-Sobolev exponent},
Nonlinear Analysis 66 (2007), 1351-1464.

\bibitem{S2} L. Barbu, C. Enache;
\emph{Maximum principles, Liouville-type theorems and
symmetry results for a general class of quasilinear anisotropic
equations}, Adv. Nonlinear Anal. 5(4) (2016), 395-405.

\bibitem{S3} M. Bidaut-Veron;
\emph{Local behaviour of the solutions of a class of nonlinear elliptic
systems}, Adv. Differential Equations, 5 (2000), 147-192.

\bibitem{S4} A. Cauchy;
\emph{M\'emoires sur les fonctions compl\'ementaires}, C. R.
Acad. Sci. Paris, 19 (1844), 1377-1384.

\bibitem{S5} C. S. Chen, H. X. Song, H. W. Yang;
\emph{Liouville type theorems for stable solutions of $p$-Laplace equation
in $\mathbb{R}^N$}, Nonlinear Anal., 160 (2017), 44-52.

\bibitem{S6} L. Damascelli, A. Farina, B. Sciunzi, Enrico Valdinoci;
\emph{Liouville results for $m$-Laplace equations of Lane-Emden-Fowler type}, Ann.
Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 26 (2009), 1099-1119.

\bibitem{S7} L. D'Ambrosio, E. Mitidieri;
\emph{Quasilinear elliptic equations with critical potentials}, 
Adv. Nonlinear Anal., 6 (2) (2017), 147-164.

\bibitem{S8} E. N. Dancer, A. Farina;
\emph{On the classification of solutions of
$-\Delta u=e^u$ on $\mathbb{R}^N$: stability outside a compact set and
applications}, Proc. Amer. Math. Soc., 137 (4) (2009), 1333-1338.

\bibitem{S9} E. N. Dancer, Y. H. Du, Z. M. Guo;
\emph{Finite Morse index solutions of an
elliptic equation with supercritical exponent}, J. Differential
Equations, 250 (2011), 3281-3310.

\bibitem{S10} E. DiBenedetto;
\emph{$C^{1+\alpha}$ local regularity of weak solutions of
degenerate elliptic equations}. Nonlinear Analysis, 7 (1983), 827-850

\bibitem{S11} Y. H. Du, Z. M. Guo;
\emph{Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic
equations}, Adv. Differential Equations, 18 (2013), 737-768.

\bibitem{S12} Y. H. Du, Z. M. Guo;
\emph{Positive solutions of an elliptic equation with negative
exponent: Stability and critical power},  J. Differential Equations
246 (2009), 2387-2414.

\bibitem{S13} A. Farina;
\emph{On the classification of solutions of Lane-Emden
equation on unbounded domains of $\mathbb{R}^N$}, J. Math. Pures Appl.,
87 (2007) 537-561.

\bibitem{S14} A. Farina;
\emph{Stable solutions of $-\Delta u=e^u$ on $\mathbb{R}^N$}, C. R. Acad. Sci.
Paris I, 345 (2007), 63-66.

\bibitem{S15} M. Fazly;
\emph{Liouville type theorems for stable solutions of certain
elliptic systems}, Adv. Nonlinear Stud. 12 (2012), 1-17.

\bibitem{S16} R. Filippucci, P. Pucci, V. R\u{a}dulescu;
\emph{Existence and non-existence results for quasilinear elliptic 
exterior problems with nonlinear boundary conditions}, 
Comm. Partial Differential Equations 33 (4-6) (2008), 706-717.

\bibitem{S17}  R. Filippucci, P. Pucci, F. Robert;
\emph{On a $p$-Laplace equation with multiple critical
nonlinearities}, Journal de Mathematiques Pures et Appliquees, 91 (2)
(2009), 156-177.

\bibitem{S18} M. Ghergu, V. R\u{a}dulescu;
\emph{Multi-parameter bifurcation and
asymptotics for the singular Lane-Emden-Fowler equation with a
convection term}, Proc. Roy. Soc. Edinburgh Sect. A
135 (1) (2005), 61-83.

\bibitem{S19} M. Ghergu, V. R\u{a}dulescu;
\emph{Singular elliptic problems with lack of compactness}, 
Annali di Matematica, 185 (2006), 63-79.

\bibitem{S20} M. Ghergu, V. R\u{a}dulescu;
\emph{Singular elliptic problems: bifurcation
and asymptotic analysis, Oxford Lecture Series in Mathematics and
its Applications}, 37. The Clarendon Press, Oxford University Press,
Oxford, 2008.

\bibitem{S21} B. Gidas, J. Spruck;
\emph{Global and local behavior of positive
solutions of nonlinear elliptic equations}. Commun. Pure Appl. Math.
34(4)(1981)525-598.

\bibitem{S22} Z. M. Guo, L. F. Mei;
\emph{Liouville type results for a $p-$Laplace equation
withnegative exponent}, Acta Mathematica Sinica, English series,
32 (12) (2016), 1515-1540.

\bibitem{S23} W. Jeong, Y. Lee;
\emph{Stable solutions and finite Morse index solutions of nonlinear
elliptic equations with Hardy potential}, Nonlinear Analysis
87 (2013), 126-145.

\bibitem{S24} J. Liouville;
\emph{Remarques de M. Liouville sur ``Construction g\'{e}om\'{e}trique 
des amplitudes   dans les fonctions elliptiques'' par M. Charles}. 
C.R. Acad. Sci. Paris. 19 (1844). 1261-1263.

\bibitem{S25} L. Ma, J. C. Wei;
\emph{Properties of positive solutions to an elliptic equation
with negative exponent}, J. Funct. Anal., 254 (2008), 1058-1087.

\bibitem{S26} Q. Phan, P. Souplet;
\emph{Liouville-type theorems and bounds of solutions of Hardy-H\'{e}non
equations}, J. Differential Equations, 252 (2012), 2544-2562.

\bibitem{S27} Phuong Le;
\emph{Liouville theorems for stable solutions of $p$-Laplace
equations with convex nonlinearities}, J. Math. Anal. Appl.
443 (2016), 431-444.

\bibitem{S28} C. Wang, D. Ye;
\emph{Some Liouville theorems for H\'{e}non type elliptic equations},
J. Funct. Anal., 262 (2012), 1705-1727.

\bibitem{S29} B. J. Xuan;
\emph{The solvability of quasilinear
Brezis-Nirenberg-type problems with singular weights}, Nonlinear
Analysis 62(2005)703-725.

\end{thebibliography}

\end{document}