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

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\inΩ, (-\triangle)^sv=|x|^bu^q,&x\inΩ, u=v=0,&x\in R^n\backslashΩ, \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.

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• Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres