  In this paper, we consider the existence and non-existence of non-trivial
solution to a Brezis-Nirenberg type problem with singular weights. First, we
obtain a compact imbedding theorem which is an extension of the classical
Rellich-Kondrachov compact imbedding theorem, and consider the corresponding
eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a
non-existence result. Thirdly, based on a generalized concentration compactness
principle, we will give some abstract conditions when the functional satisfies
the (PS)$_c$ condition. Finally, based on the explicit form of the extremal
function, we will obtain some existence results to the problem.
