Liouville type theorems for fractional elliptic problems

TL;DR

This paper proves Liouville type theorems for stable solutions of fractional elliptic equations and systems, providing a classification of solutions in the entire space, which advances understanding of fractional PDEs.

Contribution

It establishes the first classification results for stable solutions of the fractional Lane-Emden system and Liouville theorems for fractional elliptic equations with convex, nondecreasing nonlinearities.

Findings

Liouville theorems for stable solutions of fractional equations.

Classification of stable solutions to fractional Lane-Emden systems.

First such classification in the literature.

Abstract

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