Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight

Abstract

Let $p>0$ and $(-\Delta )^{s}$ is the fractional Laplacian with $0< s<1$ . The purpose of this paper is to establish a classification result for positive stable solutions to a fractional singular elliptic equation with weight $$ (-\Delta )^{s} u=-h(x)u^{-p}\text{ in }\mathbb{R}^{N}. $$ Here $N>2s$ and $h$ is a nonnegative, continuous function satisfying $h(x)\geq C|x|^{a}$ , $a\geq 0$ , when $|x|$ large. We prove the nonexistence of positive stable solutions of this equation under the condition $$ N< 2s+\frac{2(a+2s)}{p+1}\left (p+\sqrt{p^{2}+p}\right ) $$ or equivalently $$ p>p_{c}(N,s,a), $$ where $$ p_{c}(N,s,a)= \textstyle\begin{cases} \frac{(N-2s)^{2}-2(N+a)(a+2s)+2\sqrt{(a+2s)^{3}(2N-2s+a)}}{(N-2s)(10s+4a-N)}&\text{ if }N< 10s+4a \\ +\infty &\text{ if }N\geq 10s+4a \end{cases}\displaystyle . $$