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Multiple Solutions for Nonlinear Periodic Problems

  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 $C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance
MSC Classifications: 34B15, 34B18, 34C25, 58E05 show english descriptions Nonlinear boundary value problems
Positive solutions of nonlinear boundary value problems
Periodic solutions
Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
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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