Abstract and Applied Analysis

Nonhomogeneous Nonlinear Dirichlet Problems with a p -Superlinear Reaction

Leszek Gasiński and Nikolaos S. Papageorgiou

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We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reaction f ( z , ζ ) , whose primitive f ( z , ζ ) is p -superlinear near ± , but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like the p -Laplacian, the ( p , q ) -Laplacian, and the p -generalized mean curvature operator.

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Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 918271, 28 pages.

First available in Project Euclid: 5 April 2013

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Gasiński, Leszek; Papageorgiou, Nikolaos S. Nonhomogeneous Nonlinear Dirichlet Problems with a $p$ -Superlinear Reaction. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 918271, 28 pages. doi:10.1155/2012/918271. https://projecteuclid.org/euclid.aaa/1365125161

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