The authors deal with the singular variational problem S(a, b, 0) := inf u∈E,u6≡0 R RN (||x|−a∇u|m + 0|x|−(a+1)m|u|m) dx(R RN ||x|−bu|p dx)m/p
as well as eS = eS(a, b, 1, 2) := infu,v∈E,(u,v)6≡(0,0) R RN J(u, v) dx(R RN |x|−bp|u| |v| dx)m/p where J(u, v) = ||x|−a∇u|m + 1|x|−(a+1)m|u|m + ||x|−a∇v|m + 2|x|−(a+1)m|v|m,N ≥ m + 1 > 2, 0 ≤ a < N−m m , a ≤ b < a + 1 and p = p(a, b) = + =Nm N−m+m(b−a) , , ≥ 1,E = D1,m a (RN). The aim of this paper is to show the existence of minimizer for S(a, b, 0) and eS(a, b, 1, 2).
