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 λ>0, we analyze the attainability of the optimal constant μα,λ(Ω):=inf {[u]2s,Ω+λ∫Ω∣∣∣u∣∣∣2dx:u∈Hs(Ω),∫Ω|u(x)|2s,α|x|αdx=1}, where 0<s<1, n>4s, 0≤α<2s, 2s,α=2(n−α)n−2s, and Ω⊂Rn is a bounded domain such that 0∈Ω.