Symmetry and nonexistence of positive solutions for fractional systems

TL;DR

This paper investigates conditions under which positive solutions do not exist for a class of fractional Hénon systems, employing the method of moving planes to establish nonexistence in the subcritical case.

Contribution

It provides new nonexistence results for positive solutions of fractional Hénon systems using the direct method of moving planes.

Findings

Nonexistence of positive solutions in the subcritical case.

Application of the moving planes method to fractional systems.

Extension of nonexistence results to fractional Laplacian systems.

Abstract

This paper is devoted to study the nonexistence results of positive solutions for the following fractional H$\acute{e}$non system \begin{eqnarray*}\left\{ \begin{array}{lll} &(-\triangle)^{\alpha/2}u=|x|^av^p,~~~&x\in R^n, &(-\triangle)^{\alpha/2}v=|x|^bu^q,~~~ &x\in R^n, &u\geq0, v\geq 0, \end{array} \right. \end{eqnarray*} where $0<\alpha<2$, $0<p,q<\infty$, $a$, $b$ $\geq0$, $n\geq2$. Using a direct method of moving planes, we prove non-existence of positive solution in the subcritical case.