Multiple solutions for a fractional [formula omitted]-Kirchhoff problem with Hardy nonlinearity

Abstract

This paper is devoted to study the existence of multiple solutions for the following fractional p-Kirchhoff problem (0.1)M∫R2n|u(x)−u(y)|p|x−y|n+psdxdy(−△)psu=λ|u|q−2u+|u|r−2u|x|α,inΩ,u=0,inRn∖Ω,where (−△)ps denotes the fractional p-Laplace operator, Ω is a smooth bounded set in Rn containing 0 with Lipschitz boundary, M(t)=a+btθ−1 with a≥0, b>0, θ>1. λ>0, 1<q<p<θp≤r≤pα∗, pα∗=(n−α)pn−ps is the fractional critical Hardy–Sobolev exponent for 0≤α<ps<n. By using fibering maps and Nehari manifold, we obtain that the existence of multiple solutions to problem (0.1) for both Hardy–Sobolev subcritical and critical cases. In particular, the concentration compactness principle will be used to overcome the lack of compactness for the critical case.