Existence and non-existence results for quasilinear Kirchhoff problems with the Hardy–Sobolev exponent

Abstract

In this paper, we are concerned with the following quasilinear elliptic problems of Kirchhoff type: [a+b(∫RN|∇u|p−μ|u|p|x|pdx)θ−1](−Δpu−μ|u|p−2u|x|p)=|u|p∗(α)−2u|x|α+λf(x)|u|q−2u|x|β,where a,λ≥0, b>0, θ>1, 0≤μ<μ¯=[(N−p)∕p]p, α,β∈[0,p) and q∈(1,p) are constants and Δpu=div(|∇u|p−2∇u) with 1<p<N, p∗(α)=p(N−α)∕(N−p) is the Hardy–Sobolev exponent. We establish the non-existence and the existence of infinitely many nontrivial solutions when λ=0 for the above problem. Also for suitable weight function f(x), we prove the existence of two positive solutions by using Nehari manifold and fibering map when λ>0 is sufficiently small.