On A Singular Variational Problem

In this paper, we deal with the singular variational problems: S(a,b,λ₀):=(?)∫R^N(|x|^-am|â–½u|^m+λ₀|x|^-(a+1)m|u|^m)dx/∫R^N|x|^-bp|u|^pdx)^m/p,as well as (?)=(?)(b,bλ1,λ₂):=(?)∫R^NJ(v,v)dx/(∫R^N|x|^-bp|u|^α|u|^βdx)^m/p,where J(u, v)=|x|^-am|â–½u|^m+λ₁|x|⁽-(a+1)m|u|^m+|x|^-am|â–½v|^m+λ₂|x|^-(a+1)m|v|^m,N≥m+1＞2, 0≤a＜N-m/m,a≤b＜a+1 and p=p(a,b)=α+β=Nm/N-m+m(b-a),α,β≥1, E≡D_a ^1,m(R^N). The aim of this paper is to show the exis-tence of minimizer for S(a, b,λ₀) and (?)(a, b,λ₁,λ₂).