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# Multiple Solutions for Nonlinear Periodic Problems

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Published:2011-08-03
Printed: Jun 2013
• Sophia Th. Kyritsi,
Department of Mathematics, Hellenic Naval Academy, Piraeus 18539, Greece
• Nikolaos S. Papageorgiou,
National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
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## Abstract

We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term $f(t,x)$ that exhibits a $(p-1)$-superlinear growth in $x \in \mathbb{R}$ near $\pm\infty$ and near zero. A special case of the differential operator is the scalar $p$-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).
 Keywords: $C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance
 MSC Classifications: 34B15 - Nonlinear boundary value problems 34B18 - Positive solutions of nonlinear boundary value problems 34C25 - Periodic solutions 58E05 - Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)

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