Liouville type theorems for fractional elliptic problems

Abstract

In this paper, we establish Liouville type theorems for stable solutions on the whole space ${\mathbb{R}}^N$ to the fractional elliptic equation $$(-\Delta {)}^{s}u=f(u)$$ where the nonlinearity is nondecreasing and convex. We also obtain a classification of stable solutions to the fractional Lane-Emden system $$\begin{cases} (-Δ)^s u = v^p \mbox{ in }\mathbb R^N (-Δ)^s v = u^q \mbox{ in }\mathbb R^N \end{cases}$$ with $p>1$ and $q>1$. In our knowledge, this is the first classification result for stable solutions of the fractional Lane-Emden system in literature.

20 pages, comment are welcome