Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form -div(a(x)del u) = b(x)u(p) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, N >= 1, are obtained via new embeddings of some weighted Sobolev spaces with singular weights a(x) and b(x). It is seen that a(x) and b(x) admit many singular points in Omega. The main embedding results in this paper provide some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.