Critical Fractional Problems with Weights: Existence, Minimization, and Pohozaev Obstructions

Recently, a great amount of attention has been focused on the study of fractional and nonlocal operators of the elliptic type both for pure mathematical research and in view of concrete real-world applications. We are interested in proving the existence and nonexistence of solutions of a minimizing problem involving a fractional Laplacian with weight. We consider the nonlocal minimizing problem on H0s(Ω)⊂Lqs(Ω), with qs:=2nn−2s, s∈(0,1), and n≥3infu∈H0s(Ω)||u||Lqs(Ω)=1∫Rnp(x)|(−Δ)s2u(x)|2dx−λ∫Ω|u(x)|2dx, where Ω is a bounded domain in Rn, p:Rn→R is a given positive weight presenting a global positive minimum p0>0 at a∈Ω, and λ is a real constant. The objective of this paper is to show that minimizers do not exist for some k,s,λ, and n. After that, we show some nonexistence results thanks to a fractional Pohozaev identity and fractional Hardy inequality.