Some remarks on the solvability of non-local elliptic problems with the Hardy potential

Abstract

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ) s , with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝ N , N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < Λ N,s , the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with α λ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (P μ ).