A Pohozaev Identity for the Fractional H$\acute{e}$non System

TL;DR

This paper derives a Pohozaev identity for a fractional He9non-Lane-Emden system, enabling the authors to prove the nonexistence of positive solutions in certain critical regimes.

Contribution

It introduces a new Pohozaev identity for fractional Laplacian systems with He9non weights, extending previous results to fractional and weighted cases.

Findings

Nonexistence of positive solutions in critical and supercritical cases.

Extension of Pohozaev identity to fractional He9non systems.

Application to prove nonexistence results in star-shaped domains.

Abstract

In this paper, we study the Pohozaev identity associated with a H$\acute{e}$non-Lane-Emden system involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^su=|x|^av^p,&x\in\Omega, (-\triangle)^sv=|x|^bu^q,&x\in\Omega, u=v=0,&x\in R^n\backslash\Omega, \end{array} \right. \end{equation} in a star-shaped and bounded domain $\Omega$ for $s\in(0,1)$. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritical cases.