On G-symmetric solutions of a quasilinear elliptic equation involving critical Hardy–Sobolev exponent

Abstract

This paper deals with the class of singular quasilinear elliptic problem − Δ p u = μ | u | p − 2 u | x | p + k ( x ) | u | p ⁎ ( s ) − 2 u | x | s + λ f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a smooth bounded domain, and Ω is G-symmetric with respect to a subgroup G of O ( N ) , 0 ∈ Ω , 1 < p < N , Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplacian, 0 ⩽ μ < μ ¯ , μ ¯ = ( N − p p ) p , 0 ⩽ s < p , λ ⩾ 0 , p ⁎ ( s ) = ( N − s ) p N − p , k ( x ) is continuous and G-symmetric on Ω ¯ , and f : Ω × R ↦ R is a continuous nonlinearity of lower order satisfying some conditions. By using the variational methods and analytic techniques, we obtain several existence and multiplicity results of G-symmetric solutions under certain hypotheses on μ, λ and k.