The solvability of quasilinear Brezis–Nirenberg-type problems with singular weights Original Research Article

In this paper, we consider the existence and non-existence of non-trivial solutions to quasilinear Brezis–Nirenberg-type problems with singular weights. First, we shall 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 obtain a non-existence result. Thirdly, thanks to the generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)cc condition. Finally, basing on the explicit form of the extremal function, we will obtain some existence results.