Classification of positive solutions of Hardy-Sobolev equation without the finite volume constraints

Abstract

In this paper, we are concerned with the critical Hardy-Sobolev equation \begin{equation*} -Δ_{p}u = \frac{u^{p^{*}_s-1}}{|x|^{s}}, \ \ x\in \mathbb{R}^n \end{equation*} where $p_s^{\ast }=\frac{(n-s)p}{n-p}$ denotes the critical Hardy-Sobolev exponent. We classify the positive solutions of this equation for $0<s<\frac{p-1}{p}$ and $\frac{(2s+n+1)+\sqrt{(2s+n+1{)}^2-12s}}{6}\le p<n$ without finite volume constraints, which extends Ou's result in \cite{9} in the literature. The method is based on constructing suitable vector fields integral inequality and using Newton's type inequality.

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