\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 71, 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-2018/71\hfil Stable solutions to weighted quasilinear problems] {Stable solutions to weighted quasilinear problems of Lane-Emden type} \author[P. Le, V. Ho \hfil EJDE-2018/71\hfilneg] {Phuong Le, Vu Ho} \address{Phuong Le \newline Department of Mathematical Economics, Banking University of Ho Chi Minh City, Vietnam} \email{phuongl@buh.edu.vn} \address{Vu Ho (corresponding author)\newline Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam. \newline Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam} \email{hovu@tdt.edu.vn} \dedicatory{Communicated by Vicentiu Radulescu} \thanks{Submitted July 11, 2017. Published March 15, 2018.} \subjclass[2010]{35B53, 35J92, 35B08, 35B35} \keywords{Quasilinear problems; stable solutions; Lane-Emden nonlinearity; \hfill\break\indent Liouville theorems} \begin{abstract} We prove that all entire stable $W^{1,p}_{\rm loc}$ solutions of weighted quasilinear problem $$ -\operatorname{div} (w(x)|\nabla u|^{p-2} \nabla u) = f(x)|u|^{q-1}u $$ must be zero. The result holds true for $p \ge 2$ and $p-1 < q < q_c(p,N,a,b)$. Here $b > a - p$ and $q_c(p,N,a,b)$ is a new critical exponent, which is infinity in low dimension and is always larger than the classic critical one, while $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are nonnegative functions such that $w(x) \le C_1|x|^a$ and $f(x) \ge C_2|x|^b$ for large $|x|$. We also construct an example to show the sharpness of our result. \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 statement of main results} In this article we assume that $q>p-1\ge1$ and $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are nonnegative functions. Let us consider the following weighted quasilinear equation \begin{equation}\label{problem:main} -\operatorname{div} (w(x)|\nabla u|^{p-2} \nabla u) = f(x)|u|^{q-1}u \quad \text{in } \mathbb{R}^N. \end{equation} If $w\equiv1$, the left hand side of \eqref{problem:main} becomes the well-known $p$-Laplace operator. The terms $w(x)$ and $f(x)$ are usually regarded as weights while $|u|^{q-1}u$ is the so-called Lane-Emden nonlinearity. Because of the degenerate nature of the term $|\nabla u|^{p-2}$ when $p>2$, solutions to \eqref{problem:main} must be understood in the weak sense. Moreover, solutions to elliptic equations with Hardy potentials may possess singularities (see Proposition \ref{counterexample} for an example). Therefore, it is natural to study weak solutions of \eqref{problem:main} in a suitable weighted Sobolev space. For this purpose, let us define $$ \|\varphi\|_{w} = \Big(\int_{\mathbb{R}^N} w(x)|\nabla \varphi|^p \,dx \Big)^{1/p} $$ for $\varphi\in C^\infty_c(\mathbb{R}^N)$ and denote by $W^{1,p}_0(\mathbb{R}^N,w)$ the closure of $C^\infty_c(\mathbb{R}^N)$ with respect to the $\|\cdot\|_{w}$-norm. Remark that for $w\in L^1_{\rm loc}(\mathbb{R}^N)$ we have $C^1_c(\mathbb{R}^N) \subset W^{1,p}_0(\mathbb{R}^N,w)$ and $u \in W^{1,p}_{\rm loc}(\mathbb{R}^N,w)$ means that if for any $\varphi\in C^\infty_c(\mathbb{R}^N)$, there holds $u\varphi\in W^{1,p}_0(\mathbb{R}^N,w)$. Let us make also the meaning of weak solution and stable solution more precisely. \begin{definition} \label{def1.1}\rm A function $u\in W^{1,p}_{\rm loc}(\mathbb{R}^N,w)$ is said to be a \textit{weak solution} of \eqref{problem:main} if $f(x)|u|^q \in L^1_{\rm loc}(\mathbb{R}^N)$ and \begin{equation}\label{definition:solution} \int_{\mathbb{R}^N} w(x)|\nabla u|^{p-2} (\nabla u , \nabla\varphi) \, dx = \int_{\mathbb{R}^N} f(x)|u|^{q-1}u\varphi \, dx \end{equation} for all $\varphi \in C_c^1(\mathbb{R}^N)$. \end{definition} \begin{definition} \label{def1.2}\rm A weak solution $u$ of \eqref{problem:main} is \textit{stable} if \begin{equation}\label{definition:stablesolution} \int_{\mathbb{R}^N} w(x)\left[|\nabla u|^{p-2} |\nabla\varphi|^2 + (p-2)|\nabla u|^{p-4} (\nabla u , \nabla\varphi)^2\right] \,dx \ge q\int_{\mathbb{R}^N} f(x) |u|^{q-1}\varphi^2 \,dx \end{equation} for all $\varphi \in C_c^1(\mathbb{R}^N)$. \end{definition} We recall that the stability condition translates into the fact that the second variation at $u$ of the energy functional $$ E(u) = \int_{\mathbb{R}^N} \left(\frac{w(x)|\nabla u|^p}{p} - \frac{f(x)|u|^{q+1}}{q+1}\right) \, dx $$ is nonnegative. Therefore all the local minima of the functional are stable weak solutions of \eqref{problem:main}. \begin{proposition} \label{prop1.3} If $u$ is a stable solution of \eqref{problem:main}, then \begin{equation}\label{definition:stable} (p-1) \int_{\mathbb{R}^N} w(x)|\nabla u|^{p-2} |\nabla\varphi|^2 \,dx \ge q\int_{\mathbb{R}^N} f(x)|u|^{q-1}\varphi^2 \,dx \end{equation} for every $\varphi \in C_c^1(\mathbb{R}^N)$. \end{proposition} We remark that \eqref{definition:stablesolution} and \eqref{definition:stable} hold for any $\varphi\in W^{1,p}_0(\mathbb{R}^N,w)$ by density arguments. In this article we prove a Liouville type theorem for stable solutions of \eqref{problem:main}. We recall that Liouville type theorems concern about the nonexistence of nontrivial solution in the entire Euclidean space $\mathbb{R}^N$. This type of theorems for \eqref{problem:main} has drawn much attention in the last four decades. Let us mention the pioneering article \cite{BGJS81}, where Gidas and Spruck established the optimal nonexistence result for positive solutions to the equation $-\Delta u = |u|^{q-1}u \quad \text{in } \mathbb{R}^N$. They proved that this equation has no positive solution if and only if $q$ is less than the critical exponent $\frac{N+2}{N-2}$, which is $\infty$ if $N=2$. In recent years, not only weak and positive solutions but also other types of solutions to equation \eqref{problem:main} such as stable solutions have been studied immensely by several authors. Readers can find physical motivation and recent development on the topic of stable solutions in monograph \cite{LD11} by Dupaigne and references therein. We should refer to the works \cite{AF05,AF07} by Farina for Lane-Emden equation $$ -\Delta u = |u|^{q-1}u \quad \text{in } \mathbb{R}^N, $$ where he proved that all stable $C^2$ solutions must be zero if $1 p-1$ and $u$ is a stable solution of \eqref{problem:main}. Then for any \[ \alpha \in \Big(1,\frac{2q -p + 1 + 2\sqrt{q(q-p+1)}}{p-1}\Big), \] there exists a constant $C = C(p,q,\alpha) > 0$ such that for any function $\eta\in C_c^1(\mathbb{R}^N)$ with $0\le\eta\le1$ and $\nabla \eta=0$ in a neighborhood of $\{x\in\mathbb{R}^N: f(x)=0\}$ we have \begin{equation}\label{estimate_test} \begin{aligned} &\int_{\mathbb{R}^N} \left(w(x) |\nabla u|^p |u|^{\alpha-1} + f(x)|u|^{\alpha + q}\right) \eta^{\frac{p(\alpha+q)}{q-p+1}} \,dx \\ &\le C \int_{\mathbb{R}^N} w(x)^{\frac{\alpha+q}{q-p+1}} f(x)^{-\frac{\alpha+p-1}{q-p+1}}|\nabla\eta|^{\frac{p(\alpha+q)}{q-p+1}} \,dx. \end{aligned} \end{equation} \end{proposition} With the help of Proposition \ref{proposition:estimate}, it is not hard to obtain our main result. \begin{theorem}\label{theorem:instability} Let $b > a - p$ and $C_1,C_2,R_0>0$. Suppose that $w(x) \le C_1|x|^a$ and $f(x) \ge C_2|x|^b$ for a.e. $x \in \mathbb{R}^N \setminus B(0,R_0)$, in addition, $w(x)+f(x)>0$ for a.e. $x \in B(0,R_0)$. Let $u$ be a stable solution of equation \eqref{problem:main}. Assume that \begin{gather*} p-1 < q < q_c(p,N,a,b), \quad \text{if } N > \frac{(p-a)(p+3) + 4b}{p - 1},\\ p-1 < q < \infty, \quad \text{if } N \le \frac{(p-a)(p+3) + 4b}{p - 1} \end{gather*} with the critical exponent \begin{align*} q_c(p,N,a,b) &= \frac{2(p-a+b)\sqrt{(p-1)(p-a+b)(Np+bp+a-b-p)}}{(N+a-p)[(p-1)N-(p-a)(p+3)-4b]}\\ &\quad+ \Big(p-1)[N^2(p-1)-p(N(p+2)-p)+a(N(2a+p)-2p+a) \\ &\quad +b(N(p-4)-p^2+pa-2b)]\Big) \\ &\quad \div \Big((N+a-p)[(p-1)N-(p-a)(p+3)-4b]\Big). \end{align*} Then $u \equiv 0$. \end{theorem} \begin{remark}\label{remark:instability} \rm Since $b > a - p$, if $N > \frac{(p-a)(p+3) + 4b}{p - 1}$, we deduce that $a - p > -N$ and $q_c(p,N,a,b)$ is well-defined. The assumption on $q$ in Theorem \ref{theorem:instability} is equivalent to $$ N < \frac{(p - a + b)(2q -p + 1 + 2\sqrt{q(q-p+1)}) + q(p-a)(p-1) + b(p-1)^2}{(p-1)(q-p+1)}. $$ Indeed, the critical exponent $q_c(p,N,a,b)$ is explicitly computed by solving the above quadratic inequation in $q$. \end{remark} \begin{remark} \label{rmk1.7} \rm If $a=0$, then \begin{align*} q_c(p,N,0,b) &= \frac{2(p+b)\sqrt{(p-1)(p+b)(Np+bp-b-p)}}{(N-p)[(p-1)N-p(p+3)-4b]}\\ &\quad+ \frac{(p-1)[N^2(p-1)-p(N(p+2)-p)+b(N(p-4)-p^2-2b)]} {(N-p)[(p-1)N-p(p+3)-4b]}, \end{align*} which is the critical exponent $q_c$ in \cite{CC17}. Furthermore, if $a=b=0$, then we obtain \begin{align*} q_c(p,N,0,0) = \frac{2p^2\sqrt{(p-1)(N-1)} + (p-1)[N^2(p-1)-p(N(p+2)-p)]} {(N-p)[(p-1)N-p(p+3)]}, \end{align*} which equals the critical exponent $p_c$ in \cite{LDAFBSEV09}. 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 $a=0$ and $p=2$, we find \[ q_c(2,N,0,b) = \frac{2(b+2)\sqrt{(b+2)(b+2N-2)} + (N-2)^2 - 2(b+2)(b+N)}{(N-2)(N-4b-10)}, \] which is the critical exponent $\overline p(b)$ in \cite{EDYDZG11}. Finally, if $a=b=0$ and $p=2$, we have \[ q_c(2,N,0,0) = \frac{8\sqrt{N-1} + N^2-8N+4}{(N-2)(N-10)}. \] It is the critical exponent $p_c$ in \cite{AF07}. Therefore, our conclusion in Theorem \ref{theorem:instability} extends results in \cite{CC17,LDAFBSEV09,EDYDZG11,AF07} to stable solutions of class $W^{1,p}_{\rm loc}$. \end{remark} The assumption on $q$ in Theorem \ref{theorem:instability} is optimal. Indeed, let us consider the limit problem \begin{equation}\label{problem:henon} -\operatorname{div} (|x|^a|\nabla u|^{p-2} \nabla u) = |x|^b|u|^{q-1}u \quad \text{in } \mathbb{R}^N. \end{equation} We have the following result. \begin{proposition}\label{counterexample} Let $b > a - p$. Suppose that $N > \frac{(p-a)(p+3) + 4b}{p - 1}$ and $q \ge q_c(p,N,a,b)$, which is defined in Theorem \ref{theorem:instability}, then $U(x) = m/|x|^n$ is a stable solution of equation \eqref{problem:henon}. Here, $$ n=\frac{p-a+b}{q-p+1}\quad\text{and}\quad m = [n^{p-1}(N+a-1-(n+1)(p-1))]^{1/(q-p+1)}. $$ \end{proposition} \section{Proofs} This section is devoted to the proofs of Proposition \ref{proposition:estimate}, Theorem \ref{theorem:instability} and Proposition \ref{counterexample}. For convenience, we always denote by $C$ a generic constant whose concrete values may change from line to line or even in the same line. If this constant depends on an arbitrary small number $\varepsilon$, then we may denote it by $C_\varepsilon$. We also use Young inequality in the form $ab \le \varepsilon a^p + C_\varepsilon b^q$ for $p,q>0$ satisfying $\frac{1}{p} + \frac{1}{q} = 1$. \begin{proof}[Proof of Proposition \ref{proposition:estimate}] For each $k\in\mathbb{N}$ we define $$a_k(t) = \begin{cases} |t|^{\frac{\alpha-1}{2}}t, & |t| < k,\\ k^{\frac{\alpha-1}{2}}t, & |t| \ge k, \end{cases} \quad\text{and}\quad b_k(t) = \begin{cases} |t|^{\alpha-1}t, & |t| < k,\\ k^{\alpha-1}t, & |t| \ge k. \end{cases} $$ It is easy to check that \begin{equation}\label{testfuntion} \begin{gathered} a_k(t)^2 \ge t b_k(t),\quad a_k'(t)^2 \le \frac{(\alpha+1)^2}{4\alpha} b_k'(t), \\ |a_k(t)|^p a_k'(t)^{2-p} + |b_k(t)|^p b_k'(t)^{1-p} \le C |t|^{\alpha + p - 1} \end{gathered} \end{equation} for all $t\in\mathbb{R}$, where $C$ depends only on $p$ and $\alpha$. Moreover, since $u\in W^{1,p}_{\rm loc}(\mathbb{R}^N,w)$, clearly $a_k(u), b_k(u) \in W^{1,p}_{\rm loc}(\mathbb{R}^N,w)$ for any $k \in \mathbb{N}$. We split the proof into four steps. \smallskip \noindent\textbf{Step 1.} For any $\varepsilon \in (0, 1)$, any $k\in\mathbb{N}$ and any nonnegative function $\psi\in C_c^1(\mathbb{R}^N)$, there exists a constant $C_\varepsilon = C(p,\varepsilon) > 0$ such that \begin{equation}\label{s1} \begin{aligned} &(1-\varepsilon) \int_{\mathbb{R}^N} w(x) |\nabla u|^p b_k'(u) \psi^p \,dx \\ &\le C_\varepsilon\int_{\mathbb{R}^N} w(x) |b_k(u)|^p b_k'(u)^{1-p} |\nabla\psi|^p \,dx + \int_{\mathbb{R}^N} f(x) |u|^{q-1}u b_k(u) \psi^p \,dx. \end{aligned} \end{equation} To prove this, using $\varphi=b_k(u)\psi^p$ as a test function. Since $$ \nabla\varphi = b_k'(u)\psi^p\nabla u + pb_k(u)\psi^{p-1}\nabla\psi, $$ using \eqref{definition:solution} we obtain \begin{align*} &\int_{\mathbb{R}^N} w(x) |\nabla u|^p b_k'(u) \psi^p \,dx + p\int_{\mathbb{R}^N} w(x) |\nabla u|^{p-2} b_k(u) \psi^{p-1} (\nabla u, \nabla\psi) \,dx \\ &= \int_{\mathbb{R}^N} f(x) |u|^{q-1}u b_k(u) \psi^p \,dx. \end{align*} Therefore, \begin{align*} &\int_{\mathbb{R}^N} w(x) |\nabla u|^p b_k'(u) \psi^p \,dx\\ &\le p\int_{\mathbb{R}^N} w(x) |\nabla u|^{p-1} |b_k(u)| \psi^{p-1} |\nabla\psi| \,dx + \int_{\mathbb{R}^N} f(x) |u|^{q-1}u b_k(u) \psi^p \,dx\\ &\le \int_{\mathbb{R}^N} \varepsilon\left(w(x)^{\frac{p-1}{p}} |\nabla u|^{p-1} b_k'(u)^{\frac{p-1}{p}} \psi^{p-1} \right)^{\frac{p}{p-1}} \\ &\quad + C_\varepsilon\left(w(x)^{1/p}|b_k(u)|b_k'(u) ^{\frac{1-p}{p}} |\nabla\psi| \right)^p \,dx + \int_{\mathbb{R}^N} f(x) |u|^{q-1}u b_k(u) \psi^p \,dx\\ &= \varepsilon \int_{\mathbb{R}^N} w(x) |\nabla u|^p b_k'(u) \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} w(x) |b_k(u)|^p b_k'(u)^{1-p} |\nabla\psi|^p \,dx\\ &\quad + \int_{\mathbb{R}^N} f(x) |u|^{q-1}u b_k(u) \psi^p \,dx, \end{align*} which implies \eqref{s1}. \smallskip \noindent\textbf{Step 2.} For any $\varepsilon \in (0, 1)$, any $k\in\mathbb{N}$ and any nonnegative function $\psi\in C_c^1(\mathbb{R}^N)$, there exists a constant $C_\varepsilon = C(p,\varepsilon) > 0$ such that \begin{equation} \label{s2} \begin{aligned} q \int_{\mathbb{R}^N} f(x) |u|^{q-1} a_k(u)^2 \psi^p \,dx &\le \left(p-1+\varepsilon\right)\int_{\mathbb{R}^N} w(x) |\nabla u|^p a_k'(u)^2 \psi^p \,dx\\ &\quad + C_\varepsilon\int_{\mathbb{R}^N} w(x) |a_k(u)|^p a_k'(u)^{2-p} |\nabla\psi|^{p} \,dx. \end{aligned} \end{equation} To prove this, we use the stability assumption with $\varphi = a_k(u) \psi^{p/2}$. Since $$ \nabla\varphi = a_k'(u) \psi^{p/2} \nabla u + \frac{p}{2}a_k(u) \psi^{\frac{p-2}{2}} \nabla\psi, $$ using \eqref{definition:stable} we obtain \begin{equation} \label{s2.1} \begin{aligned} & q \int_{\mathbb{R}^N} f(x) |u|^{q-1} a_k(u)^2 \psi^p \,dx \\ &\le (p-1)\int_{\mathbb{R}^N} w(x) |\nabla u|^p a_k'(u)^2 \psi^p \,dx \\ &\quad + (p-1)p \int_{\mathbb{R}^N} w(x) |\nabla u|^{p-1} |a_k(u)| a_k'(u) \psi^{p-1} |\nabla\psi| \,dx \\ &\quad+ \frac{(p-1)p^2}{4}\int_{\mathbb{R}^N} w(x) |\nabla u|^{p-2} a_k(u)^2 \psi^{p-2} |\nabla\psi|^2 \,dx. \end{aligned} \end{equation} Now we use Young inequality to estimate the last two terms \begin{align*} &(p-1)p \int_{\mathbb{R}^N} w(x) |\nabla u|^{p-1} |a_k(u)| a_k'(u) \psi^{p-1} |\nabla\psi| \,dx\\ &\le \int_{\mathbb{R}^N} \frac{\varepsilon}{2}\left(w(x)^{\frac{p-1}{p}} |\nabla u|^{p-1} a_k'(u)^{\frac{2(p-1)}{p}} \psi^{p-1}\right)^{\frac{p}{p-1}} \\ &\quad + C_\varepsilon\left(w(x)^{1/p} |a_k(u)| a_k'(u)^{\frac{2-p}{p}} |\nabla\psi|\right)^{p} \,dx\\ &= \frac{\varepsilon}{2} \int_{\mathbb{R}^N} w(x) |\nabla u|^p a_k'(u)^2 \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} w(x) |a_k(u)|^p a_k'(u)^{2-p} |\nabla\psi|^{p} \,dx \end{align*} and \begin{align*} &\frac{(p-1)p^2}{4}\int_{\mathbb{R}^N} w(x) |\nabla u|^{p-2} a_k(u)^2 \psi^{p-2} |\nabla\psi|^2 \,dx\\ &\le \int_{\mathbb{R}^N} \frac{\varepsilon}{2}\left(w(x)^{\frac{p-2}{p}} |\nabla u|^{p-2} a_k'(u)^{\frac{2(p-2)}{p}} \psi^{p-2}\right)^{\frac{p}{p-2}}\\ &\quad + C_\varepsilon\left(w(x)^{\frac{2}{p}} a_k(u)^2a_k'(u)^{\frac{2(2-p)}{p}} |\nabla\psi|^2\right)^{p/2} \,dx\\ &= \frac{\varepsilon}{2} \int_{\mathbb{R}^N} w(x) |\nabla u|^p a_k'(u)^2 \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} w(x) |a_k(u)|^p a_k'(u)^{2-p} |\nabla\psi|^{p}\,dx. \end{align*} Using these two estimates into \eqref{s2.1}, we obtain \eqref{s2}. \smallskip \noindent\textbf{Step 3.} We claim that there exists a constant $C = C(p,q,\alpha) > 0$ such that for any nonnegative function $\psi\in C_c^1(\mathbb{R}^N)$ we have \begin{equation}\label{s3} \int_{\mathbb{R}^N} \left(w(x) |\nabla u|^p |u|^{\alpha - 1} + f(x) |u|^{\alpha + q}\right)\psi^p \,dx \le C\int_{\mathbb{R}^N} w(x) |u|^{\alpha + p -1} |\nabla\psi|^{p} \,dx. \end{equation} To prove this, we set $\beta_\varepsilon = 1 - \frac{(p-1+\varepsilon) (\alpha+1)^2}{4(1-\varepsilon)\alpha q}$. Since $\lim_{\varepsilon\to 0^+}\beta_\varepsilon = 1 - \frac{(p-1)(\alpha+1)^2}{4\alpha q} > 0$, we can fix some $\varepsilon\in(0, 1)$ depending on $p$, $q$ and $\alpha$ such that $\beta_\varepsilon > 0$. Collecting \eqref{s1}, \eqref{s2} and with the help of \eqref{testfuntion} we obtain \begin{align*} &q \int_{\mathbb{R}^N} f(x) |u|^{q-1} a_k(u)^2 \psi^p \,dx\\ &\le \left(p-1+\varepsilon\right)\int_{\mathbb{R}^N} w(x) |\nabla u|^p a_k'(u)^2 \psi^p \,dx \\ &\quad + C_\varepsilon\int_{\mathbb{R}^N} w(x) |a_k(u)|^p a_k'(u)^{2-p} |\nabla\psi|^{p} \,dx\\ &\le \frac{(p-1+\varepsilon)(\alpha+1)^2}{4\alpha}\int_{\mathbb{R}^N} w(x) |\nabla u|^p b_k'(u) \psi^p \,dx \\ &\quad + C_\varepsilon\int_{\mathbb{R}^N} w(x) |a_k(u)|^p a_k'(u)^{2-p} |\nabla\psi|^{p} \,dx\\ &\le \frac{(p-1+\varepsilon)(\alpha+1)^2}{4(1-\varepsilon)\alpha} \int_{\mathbb{R}^N} f(x) |u|^{q-1}u b_k(u) \psi^p \,dx\\ &\quad + C_\varepsilon\int_{\mathbb{R}^N} w(x) \left[|a_k(u)|^p a_k'(u)^{2-p} + |b_k(u)|^p b_k'(u)^{1-p}\right] |\nabla\psi|^{p} \,dx\\ &\le \frac{(p-1+\varepsilon)(\alpha+1)^2}{4(1-\varepsilon)\alpha} \int_{\mathbb{R}^N} f(x) |u|^{q-1}a_k(u)^2 \psi^p \,dx \\ &\quad + C_\varepsilon\int_{\mathbb{R}^N} w(x) |u|^{\alpha+p-1} |\nabla\psi|^{p} \,dx. \end{align*} Therefore, $$ q\beta_\varepsilon \int_{\mathbb{R}^N} f(x) |u|^{q-1}a_k(u)^2 \psi^p \,dx \le C_\varepsilon\int_{\mathbb{R}^N} w(x) |u|^{\alpha+p-1} |\nabla\psi|^{p}. $$ Letting $k\to\infty$, by the monotone convergence theorem we obtain \begin{equation}\label{s31} \int_{\mathbb{R}^N} f(x) |u|^{\alpha+q} \psi^p \,dx \le C \int_{\mathbb{R}^N} w(x) |u|^{\alpha+p-1} |\nabla\psi|^{p}, \end{equation} where $C$ depends only on $p$, $q$ and $\alpha$. On the other hand, applying \eqref{s1} for $\varepsilon=1/2$, \begin{align*} &\int_{\mathbb{R}^N} w(x) |\nabla u|^p b_k'(u) \psi^p \,dx \\ &\le C\int_{\mathbb{R}^N} w(x) |b_k(u)|^p b_k'(u)^{1-p} |\nabla\psi|^p \,dx + 2\int_{\mathbb{R}^N} f(x) |u|^{q-1}u b_k(u) \psi^p \,dx\\ &\le C\int_{\mathbb{R}^N} w(x) |u|^{\alpha+p-1} |\nabla\psi|^p \,dx + 2 \int_{\mathbb{R}^N} f(x) |u|^{q-1}a_k(u)^2 \psi^p \,dx. \end{align*} Letting $k\to\infty$, by the monotone convergence theorem and \eqref{s31} we obtain \begin{equation}\label{s32} \int_{\mathbb{R}^N} w(x) |\nabla u|^p |u|^{\alpha - 1} \psi^p \,dx \le C\int_{\mathbb{R}^N} w(x) |u|^{\alpha+p-1} |\nabla\psi|^p \,dx. \end{equation} Combining \eqref{s31} and \eqref{s32} we obtain \eqref{s3}. \smallskip \noindent\textbf{Step 4.} We are now in a position to prove a priori estimate \eqref{estimate_test}. Applying \eqref{s3} for $\psi=\eta^{\frac{\alpha+q}{q-p+1}}$ to obtain \begin{align*} &\int_{\mathbb{R}^N} \left(w(x) |\nabla u|^p |u|^{\alpha - 1} + f(x) |u|^{\alpha + q}\right)\eta^{\frac{p(\alpha+q)}{q-p+1}} \,dx\\ &\le C\int_{\mathbb{R}^N} w(x) |u|^{\alpha + p -1} |\nabla\eta|^{p} \eta^{\frac{p(\alpha+p-1)}{q-p+1}} \,dx\\ &\le \int_{\mathbb{R}^N} \frac{1}{2} \left(f(x)^{\frac{\alpha+p-1}{\alpha+q}} |u|^{\alpha + p -1} \eta^{\frac{p(\alpha+p-1)}{q-p+1}} \right)^{\frac{\alpha+q}{\alpha+p-1}} \\ &\quad + C \left(w(x)f(x)^{-\frac{\alpha+p-1}{\alpha+q}} |\nabla\eta|^p \right)^{\frac{\alpha+q}{q-p+1}} \,dx\\ &\le \frac{1}{2} \int_{\mathbb{R}^N} f(x) |u|^{\alpha + q} \eta^{\frac{p(\alpha+q)}{q-p+1}} \,dx + C \int_{\mathbb{R}^N} w(x)^{\frac{\alpha+q}{q-p+1}} f(x)^{-\frac{\alpha+p-1}{q-p+1}} |\nabla\eta|^{\frac{p(\alpha+q)}{q-p+1}} \,dx. \end{align*} Hence, \eqref{estimate_test} follows. \end{proof} \begin{proof}[Proof of Theorem \ref{theorem:instability}] Applying \eqref{estimate_test} for a test function $\eta_R\in C_c^1(\mathbb{R}^N)$ satisfying $0\le\eta_R\le1$ in $\mathbb{R}^N$ and \begin{gather*} \eta_R=1 \quad \text{in } B(0,R),\\ \eta_R=0 \quad \text{in } \mathbb{R}^N \setminus B(0,2R),\\ |\nabla\eta_R|\le\frac{C}{R} \quad \text{in } B(0,2R) \setminus B(0,R). \end{gather*} Consequently, for all $R>R_0$ there exists a constant $C$ independent of $R$ such that \begin{equation}\label{estimate_bound} \int_{B(0,R)} \left(w(x) |\nabla u|^p |u|^{\alpha-1} + f(x)|u|^{\alpha + q}\right) \,dx \le C R^{\theta}, \end{equation} where $$ \theta = N - \frac{(p-a)(\alpha+q) + b(\alpha+p-1)}{q-p+1}. $$ Note that $\alpha \in (1,\alpha_0(q))$ where $$ \alpha_0(t) = \frac{2t -p + 1 + 2\sqrt{t(t-p+1)}}{p-1}. $$ Let us define the function $$ g(t) = \frac{(p-a)(\alpha_0(t)+t) + b(\alpha_0(t)+p-1)}{t-p+1},\quad \text{for } t > p - 1. $$ Since $$ g'(t) = \frac{p-a+b}{(t-p+1)^2}\Big(-p-\sqrt{\frac{t-p+1}{t}}\Big) < 0, $$ the function $g(t)$ is decreasing in $t > p - 1$. On the other hand, $$ \lim_{t\to(p-1)^+} g(t) = +\infty,\quad \lim_{t\to+\infty} g(t) = \frac{(p-a)(p+3) + 4b}{p - 1}. $$ Therefore, if $N \le \frac{(p-a)(p+3) + 4b}{p - 1}$, then $N < g(q)$ since $q > p - 1$. Hence if we fix $\alpha \in [1, \alpha_0(q))$, suitably near $\alpha_0(q)$, we obtain $$ N < \frac{(p-a)(\alpha+q) + b(\alpha+p-1)}{q-p+1}, $$ which means that $\theta<0$. Then the desired result follows by letting $R \to \infty$ in \eqref{estimate_bound}. Assume now $N > \frac{(p-a)(p+3) + 4b}{p - 1}$. Since $g$ is decreasing, we obtain in this case a critical value $q_c(p, N, a, b)$ such that $N < g(q)$ for $1 < q < q_c(p, N, a, b)$. From this, the desired result follows again by letting $R \to \infty$ in \eqref{estimate_bound}. Clearly, $q_c(p, N, a, b)$ may be deduced from the equation $N = g(q)$, which is given the value in Theorem \ref{theorem:instability} (see also Remark \ref{remark:instability}). Then we complete the proof. \end{proof} \begin{proof}[Proof of Proposition \ref{counterexample}] Direct calculation yields that $U$ is a weak solution of \eqref{problem:henon}. In order to show that $U$ is stable, we need the following inequality (see \cite{LCRKLN84,FCZW01}). \begin{lemma}[Caffarelli-Kohn-Nirenberg inequality] Let $r<\frac{N-2}{2}$, then for all $\varphi\in C^1_c(\mathbb{R}^N)$ we have \begin{equation}\label{ineqcaff} \int_{\mathbb{R}^N} \frac{|\nabla\varphi|^2}{|x|^{2r}} \,dx \ge \Big( \frac{N-2-2r}{2} \Big)^{2} \int_{\mathbb{R}^N} \frac{|\varphi|^2}{|x|^{2r+2}} \,dx. \end{equation} \end{lemma} Applying \eqref{ineqcaff} with $r=\frac{(n+1)(p-2)-a}{2}$ we obtain \begin{equation}\label{ineqcaff2} \int_{\mathbb{R}^N} \frac{|\nabla\varphi|^2}{|x|^{(n+1)(p-2)-a}} \,dx \ge \Big(\frac{N-2-(n+1)(p-2)+a}{2}\Big)^2 \int_{\mathbb{R}^N} \frac{\varphi^2}{|x|^{n(q-1)-b}} \,dx. \end{equation} Since $U$ is radially symmetric and decreasing in $|x|$, by arguing as in \cite[Remark 1.7]{XCACMS09} it is necessary to check stability of $U$ for all radially symmetric test function $\varphi \in C_c^1(\mathbb{R}^N)$. For such $\varphi$ we have \begin{align*} &\int_{\mathbb{R}^N} \left[|x|^a |\nabla U|^{p-2} |\nabla\varphi|^2 + (p-2)|x|^a |\nabla U|^{p-4} (\nabla U , \nabla\varphi)^2 - q |x|^b |U|^{q-1}\varphi^2\right] \,dx\\ &= \int_{\mathbb{R}^N} \left[(p-1)|x|^a |\nabla U|^{p-2} |\nabla\varphi|^2 - q |x|^b |U|^{q-1}\varphi^2\right] \,dx\\ &= \int_{\mathbb{R}^N} \Big[(mn)^{p-2}(p-1)\frac{|\nabla\varphi|^2} {|x|^{(n+1)(p-2)-a}} - q m^{q-1} \frac{\varphi^2}{|x|^{n(q-1)-b}}\Big] \,dx\\ &\ge \int_{\mathbb{R}^N} \Big[(mn)^{p-2}(p-1) \left(\frac{N-2-(n+1)(p-2)+a}{2}\right)^2 - q m^{q-1}\Big] \frac{\varphi^2}{|x|^{n(q-1)-b}} \,dx, \end{align*} where we have used \eqref{ineqcaff2} in the last estimate. Direct computation yields \begin{align*} &(mn)^{p-2} (p-1) \Big(\frac{N-2-(n+1)(p-2)+a}{2}\Big)^2 - q m^{q-1}\\ &= (mn)^{p-2} \Big[(p-1)\left(\frac{N-2-(n+1)(p-2)+a}{2}\right)^2 \\ &- nq\left(N+a-1-(n+1)(p-1)\right)\Big]. \end{align*} We want to show that $$ (p-1)\Big(\frac{N-2-(n+1)(p-2)+a}{2}\Big)^2 - nq\left[N+a-1-(n+1)(p-1)\right] \ge 0. $$ After substituting $n=\frac{p-a+b}{q-p+1}$, this inequality is equivalent to $$ N \ge \frac{(p - a + b)(2q -p + 1 + 2\sqrt{q(q-p+1)}) + q(p-a)(p-1) + b(p-1)^2}{(p-1)(q-p+1)}. $$ The last inequality is verified by Remark \ref{remark:instability} and assumption $q \ge q_c(p,N,a,b)$. Thus, $U$ is stable. \end{proof} \begin{thebibliography}{10} \bibitem {LAEM17} L. D'Ambrosio, E. Mitidieri; \textit{Quasilinear elliptic equations with critical potentials}, Adv. Nonlinear Anal. \textbf{6} (2017), 147--164. \bibitem {XCACMS09} X. Cabr\'e, A. Capella, M. Sanch\'on; \textit{Regularity of radial minimizers of reaction equations involving the $p$-Laplacian}, Calc. Var. Partial Differential Equations \textbf{34} (2009), 475--494. \bibitem {LCRKLN84} L. Caffarelli, R. Kohn, L. Nirenberg; \textit{First order interpolation inequalities with weights}, Compositio Math. \textbf{53} (1984), 259--275. \bibitem {DCPEBS09} D. Castorina, P. Esposito, B. Sciunzi; \textit{Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$}, Commun. Pure Appl. Anal. \textbf{8} (2009), 1779--1793. \bibitem {FCZW01} F. Catrina, Z.Q. Wang; \textit{On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions}, Comm. Pure Appl. Math. \textbf{54} (2001), 229--258. \bibitem {CC17} C. Chen; \textit{Liouville type theorem for stable solutions of p-Laplace equation in $\mathbb{R}^N$}, Appl. Math. Lett. \textbf{68} (2017), 62--67. \bibitem {CCMF12} C. Cowan, M. Fazly; \textit{On stable entire solutions of semi-linear elliptic equations with weights}, Proc. Amer. Math. Soc. \textbf{140} (2012), 2003--2012. \bibitem {LDAFBSEV09} L. Damascelli, A. Farina, B. Sciunzi, E. Valdinoci; \textit{Liouville results for $m$-Laplace equations of Lane-Emden-Fowler type}, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'eaire \textbf{26} (2009), 1099--1119. \bibitem {EDYDZG11} E. N. Dancer, Y. Du, Z. M. Guo; \textit{Finite Morse index solutions of an elliptic equation with supercritical exponent}, J. Differential Equations \textbf{250} (2011) 3281--3310. \bibitem {LD11} L. Dupaigne; \textit{Stable solutions of elliptic partial differential equations}, Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics \textbf{143}, Boca Raton, FL, 2011. \bibitem {LDAF10} L. Dupaigne, A. Farina; \textit{Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$}, J. Eur. Math. Soc. \textbf{12} (2010), 855--882. \bibitem {LDMGVR07} L. Dupaigne, M. Ghergu, V. R\u{a}dulescu; \textit{Lane-Emden-Fowler equations with convection and singular potential}, J. Math. Pures Appl. \textbf{87} (2007), 563--581. \bibitem {AF05} A. Farina; \textit{Liouville-type results for solutions of $-{\Delta} u=|u|^{p-1}u$ on unbounded domains of $\mathbb{R}^N$}, C. R. Math. Acad. Sci. Paris, Ser. I \textbf{341} (2005), 415--418. \bibitem {AF07} A. Farina; \textit{On the classification of solutions of the {L}ane-{E}mden equation on unbounded domains of $\mathbb{R}^N$}, J. Math. Pures Appl., \textbf{87} (2007), 537--561. \bibitem {RFPPVR08} R. Filippucci, P. Pucci, V. R\u{a}dulescu; \textit{Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions}, Comm. Partial Differential Equations \textbf{33} (2008), 706--717. \bibitem {RFPPFR09} R. Filippucci, P. Pucci, F. Robert; \textit{On a $p$-Laplace equation with multiple critical nonlinearities}, J. Math. Pures Appl. \textbf{91} (2009), 156--177. \bibitem {MGVR06} M. Ghergu, V. R\u{a}dulescu; \textit{Singular elliptic problems with lack of compactness}, Ann. Mat. Pura Appl. \textbf{185} (2006), 63--79. \bibitem {MGVR08} M. Ghergu, V. R\u{a}dulescu; \textit{Singular Elliptic Problems: Bifurcation and Asymptotic Analysis}, Oxford Lecture Series in Mathematics and its Applications \textbf{37}. The Clarendon Press, Oxford University Press, Oxford, 2008. \bibitem {BGJS81} B. Gidas, J. Spruck; \textit{Global and local behavior of positive solutions of nonlinear elliptic equations}, Comm. Pure Appl. Math. \textbf{34} (1981), 525--598. \bibitem {WJYL13} W. Jeong, Y. Lee; \textit{Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential}, Nonlinear Anal. \textbf{87} (2013), 126--145. \bibitem {PL16-2} P. Le; \textit{Nonexistence of stable solutions to $p$-Laplace equations with exponential nonlinearity}, Electron. J. Differential Equations, \textbf{2016} (2016), No. 326, 1--5. \bibitem {PL16} P. Le; \textit{Liouville theorems for stable solutions of $p$-Laplace equations with convex nonlinearities}, J. Math. Anal. Appl. \textbf{443} (2016), 431--444. \bibitem {PLHNTN17} P. Le, H.T. Nguyen, T. Y. Nguyen; \textit{On positive stable solutions to weighted quasilinear problems with negative exponent}, Complex Var. Elliptic Equ. (2017), DOI: 10.1080/17476933.2017.1403429. \bibitem {BM17} B. Maultsby; \textit{Uniqueness of solutions to singular $p$-Laplacian equations with subcritical nonlinearity}, Adv. Nonlinear Anal. \textbf{6} (2017), 37--59. \bibitem {MNSV17} M. A. Navarro, S. Villegas; \textit{Semi-stable radial solutions of p-Laplace equations in $\mathbb{R}^N$}, Nonlinear Anal. \textbf{149} (2017), 111--116. \bibitem {CWDY12} C. Wang, D. Ye; \textit{Some Liouville theorems for H\'{e}non type elliptic equations}, J. Funct. Anal. \textbf{262} (2012), 1705--1727. \end{thebibliography} \end{document}