On the Classification of Stable Solutions of the Fractional Equation

In this paper, we study the nonexistence result for the following nonlinear elliptic equation: (−Δ)su+λu=|u|p−1uinℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-{\Delta})^{s} u+\lambda u= |u|^{p-1}u \; \;\text{ in} \; \mathbb{R}^{n} $$\end{document} where n ≥ 2s, 0 < s < 1, λ > 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.