A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian

TL;DR

This paper proves a Liouville type theorem for fractional Lane-Emden systems, extending the understanding of solutions' behavior for these equations involving the fractional Laplacian in certain parameter ranges.

Contribution

It introduces a new Liouville theorem for fractional Lane-Emden systems using the local realization of the fractional Laplacian and advanced symmetry methods.

Findings

Liouville theorem established for fractional Lane-Emden systems

Method employs local realization of fractional Laplacian as Dirichlet-to-Neumann map

Uses monotonicity, moving planes, and maximum principles in the proof

Abstract

We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=v^q&{\rm in}\,\,\R^N,\\ (-\Delta)^\alpha v=u^p&{\rm in}\,\,\R^N, \end{array} \right. \end{eqnarray*} where $ \alpha\in(0,1) $, $ N>2\alpha $ and $ p,q $ are positive real numbers and in an appropriate new range. To prove our result we will use the local realization of fractional Laplacian, which can be constructed as Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre \cite{CS}. Our proof is based on a monotonicity argument for suitable transformed functions and the method of moving planes in an infinity half cylinder based on some maximum principles which obtained by some barrier functions and a coupling argument using fractional Sobolev trace inequality.