Existence and non-existence results for quasilinear Kirchhoff problems with the Hardy–Sobolev exponent. (15th October 2018) - Shen, Liejun
Abstract: In this paper, we are concerned with the following quasilinear elliptic problems of Kirchhoff type: [ a + b ( ∫ R N | ∇ u | p − μ | u | p | x | p d x ) θ − 1 ] ( − Δ p u − μ | u | p − 2 u | x | p ) = | u | p ∗ ( α ) − 2 u | x | α + λ f ( x ) | u | q − 2 u | 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 Δ p u = 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.