Existence and multiplicity of symmetric solutions for a class of singular elliptic problems

This paper deals with the singular elliptic problem −div(|x|β∇u)=Q(x)|x|α|u|p(β,α)−2u+h(x,u)in Ω,u=0on ∂Ω, where Ω⊂RN(N≥3)Ω⊂RN(N≥3) is a smooth bounded domain, 0∈Ω0∈Ω and ΩΩ is GG-symmetric with respect to a subgroup GG of O(N)O(N), β≤0β≤0, N+β−2>0N+β−2>0, N+α>0N+α>0, α+2>βα+2>β, β≥2αp(β,α), p(β,α)=2(N+α)N+β−2, Q(x)Q(x) is continuous and GG-symmetric on Ω¯ and h:Ω×R↦Rh:Ω×R↦R is a continuous nonlinearity of lower order satisfying some conditions. Based upon the symmetric criticality principle of Palais and variational methods, we obtain several existence and multiplicity results of GG-symmetric solutions under some assumptions on QQ and hh.