Three nontrivial solutions for periodic problems with the p-Laplacian and a p-superlinear nonlinearity

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July 2009, 8(4): 1421-1437. doi: 10.3934/cpaa.2009.8.1421

Three nontrivial solutions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity

Leszek Gasiński ^1, and Nikolaos S. Papageorgiou ^2,

1.	Jagiellonian University, Institute of Computer Science, ul. Nawojki 11, 30072 Kraków
2.	Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received May 2008 Revised October 2008 Published March 2009

Abstract
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We consider a nonlinear periodic problem driven by the scalar $p$-Laplacian and a nonlinearity that exhibits a $p$-superlinear growth near $\pm\infty$, but need not satisfy the Ambrosetti-Rabinowitz condition. Using minimax methods, truncations techniques and Morse theory, we show that the problem has at least three nontrivial solutions, two of which are of fixed sign.

Keywords: Poincaré-Hopf formula., mountain pass theorem, Morse theory, Scalar p-Laplacian, critical groups.

Mathematics Subject Classification: Primary: 34B15; Secondary: 34C2.

Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Three nontrivial solutions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1421-1437. doi: 10.3934/cpaa.2009.8.1421

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