Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

Abstract

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: −()a+b∫normalΩfalse|∇ufalse|2dxnormalΔu=u5−2sfalse|xfalse|s+λfalse|xfalse|βuγ,x∈normalΩ,u>0,13.7emx∈normalΩ,u=0,13.8emx∈∂normalΩ, where normalΩ⊂ℝ3 is a bounded domain with smooth boundary ∂Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and 0≤β<5+γ2. By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.

A uniqueness positive solutions for strong singular Kirchhoff-type fractional Laplacian problems with Hardy term

“…The singular Kirchhoff type problem was firstly investigated by Liu and Sun [3], they obtain two positive solutions with the help of the Nehari method. Li, Tang and Liao in [2] and [3] Ω is nonzero and nonnegative, Liao in [4] explore the two solutions of a class of singular Kirchhoff-type problems with Hardy-Sobolev critical exponent II * . Concerning the existence theorems for entire solutions of stationary Kirchhoff Fractional p-Laplace equation with Hardy term, see [5].…”