Nonexistence of Positive Solutions for an Integral Equation Related to the Hardy-Sobolev Inequality

Let α and s be real numbers satisfying 0<s<α<n. We are concerned with the integral equation $$u(x)=\int_{R^n}\frac{u^p(y)}{|x-y|^{n-\alpha}|y|^s}dy, $$ where \(\frac{n-s}{n-\alpha}< p< \alpha^{*}(s)-1\) with \(\alpha^{*}(s)=\frac{2(n-s)}{n-\alpha}\) . We prove the nonexistence of positive solutions for the equation and establish the equivalence between the above integral equation and the following partial differential equation $$\begin{aligned} (-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^{-s}u^p. \end{aligned}$$