Communications on Pure and Applied Analysis (CPAA)

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

Pages: 1421 - 1437, Volume 8, Issue 4, July 2009      doi:10.3934/cpaa.2009.8.1421

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Leszek Gasiński - Jagiellonian University, Institute of Computer Science, ul. Nawojki 11, 30072 Kraków, Poland (email)
Nikolaos S. Papageorgiou - Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece (email)

Abstract: 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:  Scalar p-Laplacian, mountain pass theorem, critical groups, Morse theory, Poincaré-Hopf formula.
Mathematics Subject Classification:  Primary: 34B15; Secondary: 34C25.

Received: May 2008;      Revised: October 2008;      Available Online: March 2009.