Liouville Results for Fractional Elliptic Equations with Hardy Potential

Abstract

In this paper, we consider the nonexistence result for the following nonlinear elliptic equation with a Hardy term for fractional Laplacian : $$\begin{aligned} (-\Delta )^{s} u+\frac{\lambda }{|x|^{2s}} u= |x|^a |u|^{p-1}u \; \; \text{ in } \; {\mathbb {R}}^{n} \end{aligned}$$ where $$n\ge 2s$$ , $$0<s<1$$ , $$a> -2s$$ , $$\lambda >0$$ and $$p>1$$ . We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.