A minimization problem involving a fractional Hardy–Sobolev type inequality

Source 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 μ α , λ ( Ω ) : = inf { [ u ] s , Ω 2 + λ ∫ Ω | u | 2 d x : u ∈ H s ( Ω ) , ∫ Ω | u ( x ) | 2 s , α | x | α d x = 1 } , where 0 < s < 1 , n > 4 s , 0 ≤ α < 2 s , 2 s , α = 2 ( n − α ) n − 2 s , and Ω ⊂ R n is a bounded domain such that 0 ∈ Ω .