Abstract and Applied Analysis
Volume 2012 (2012), Article ID 918271, 28 pages
http://dx.doi.org/10.1155/2012/918271
Research Article

Nonhomogeneous Nonlinear Dirichlet Problems with a 𝑝 -Superlinear Reaction

Leszek Gasiński1 and Nikolaos S. Papageorgiou2

1Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza Street 6, 30-348 Kraków, Poland
2Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece

Received 9 January 2012; Revised 4 February 2012; Accepted 4 February 2012

Academic Editor: Kanishka Perera

Copyright © 2012 Leszek Gasiński and Nikolaos S. Papageorgiou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.