Existence and multiplicity results for the fractional p-Laplacian equation with Hardy-Sobolev exponents

Abstract

In this paper, we investigate the following fractional p -Laplacian problem ⎨⎩ (−Δ)pu = λ |u|p−2u+ |u| ps,α−2u |x|α in Ω, u = 0 on ∂Ω, where Ω is a bounded domain containing the origin in RN with Lipschitz boundary, p ∈ (1,∞) , s ∈ (0,1) , 0 α < ps < N and p∗s,α = (N −α)p/(N − ps) is the fractional Hardy-Sobolev exponent. We prove the existence, multiplicity and bifurcation results for the above problem. Our results extend some results in the literature for the fractional p -Laplacian problem involving critical Sobolev exponent and the p -Laplacian problem involving Hardy-Sobolev exponents.