Critical singular problems via concentration-compactness lemma

• Published in: Journal of mathematical analysis and applications 2007-02, Vol.326 (1), p.137-154
• Main Authors: Assunção, Ronaldo B., Carrião, Paulo Cesar, Miyagaki, Olimpio Hiroshi
• Format: Article
• Language: English
• Subjects: Calculus of variations and optimal control; Compactness-concentration; Degenerate quasilinear equation; Exact sciences and technology; Mathematical analysis; Mathematics; p-Laplacian; Partial differential equations; Sciences and techniques of general use; Variational methods
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2006.03.002

In this work we consider existence and multiplicity results of nontrivial solutions for a class of quasilinear degenerate elliptic equations in R N of the form (P) − div [ | x | − a p | ∇ u | p − 2 ∇ u ] + λ | x | − ( a + 1 ) p | u | p − 2 u = | x | − b q | u | q − 2 u + f , where x ∈ R N , 1 < p < N , q = q ( a , b ) ≡ N p / [ N − p ( a + 1 − b ) ] , λ is a parameter, 0 ⩽ a < ( N − p ) / p , a ⩽ b ⩽ a + 1 , and f ∈ ( L b q ( R N ) ) ∗ . We look for solutions of problem (P) in the Sobolev space D a 1 , p ( R N ) and we prove a version of a concentration-compactness lemma due to Lions. Combining this result with the Ekeland's variational principle and the mountain-pass theorem, we obtain existence and multiplicity results.