A Liouville theorem for non local elliptic equations

TL;DR

This paper establishes a Liouville-type theorem for bounded stable solutions of fractional elliptic equations involving the fractional Laplacian, extending classical results to nonlocal operators with broad applicability.

Contribution

It proves a Liouville theorem for stable solutions of fractional elliptic equations with general nonnegative nonlinearities, a significant extension of classical results to nonlocal operators.

Findings

Liouville theorem holds for bounded stable solutions of fractional elliptic equations.

The result applies to equations with any nonnegative nonlinearity.

Extends classical Liouville results to nonlocal fractional operators.

Abstract

We prove a Liouville-type theorem for bounded stable solutions $v \in C^2(\R^n)$ of elliptic equations of the type (-\Delta)^s v= f(v)\qquad {in $\R^n$,} where $s \in (0,1)$ {and $f$ is any nonnegative function}. The operator $(-\Delta)^s$ stands for the fractional Laplacian, a pseudo-differential operator of symbol $|\xi |^{2s}$.