A minimization problem involving a fractional Hardy–Sobolev type inequality

Abstract

In this work, we obtain existence of nontrivial solutions to a minimization problem involving a fractional Hardy–Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda >0$, we analyze the attainability of the optimal constant $${\mu }_{\alpha ,\lambda }(\Omega ):=inf\hspace{0.17em}\left\{\right[u{]}_{s,\Omega }^2+\lambda {\int }_{\Omega }|u{|}^{2}dx:u\in H^s(\Omega ),{\int }_{\Omega }\frac{\left|u\right(x){|}^{2_{s,\alpha }}}{|x{|}^{\alpha }}dx=1\},$$ where $0<s<1$, $n>4s$, $0\le \alpha <2s$, $2_{s,\alpha }=\frac{2(n-\alpha )}{n-2s}$, and $\Omega \subset {\mathbb{R}}^n$ is a bounded domain such that $0\in \Omega$.