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Zbl 1130.35061
Xuan, Benjin
The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. (English)
[J] Nonlinear Anal., Theory Methods Appl. 62, No. 4, A, 703-725 (2005). ISSN 0362-546X

Summary: 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 Pokhozhaev-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)$_c$ condition. Finally, basing on the explicit form of the extremal function, we will obtain some existence results.

MSC 2000:: *35J60 Nonlinear elliptic equations

Keywords: Brezis-Nirenberg problem; Singular weights; Pokhozhaev-type identity; (PS)$_c$ condition

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Zentralblatt MATH Berlin [Germany]