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Solutions and multiple solutions for superlinear perturbations of the periodic scalar p-Laplacian

Part of: Boundary value problems Variational problems in infinite-dimensional spaces Qualitative theory

Published online by Cambridge University Press:  28 June 2013

Sophia Th. Kyritsi ,
Donal O'Regan  and
Nikolaos S. Papageorgiou
Sophia Th. Kyritsi
Affiliation:
Department of Mathematics, Hellenic Naval Academy, Piraeus 18539, Greece ([email protected])
Donal O'Regan
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland ([email protected])
Nikolaos S. Papageorgiou
Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece ([email protected])
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Abstract

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We consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a reaction term which exhibits a (p – 1)-superlinear growth near ±∞ but need not satisfy the Ambrosetti-Rabinowitz condition. Combining critical point theory with Morse theory we prove an existence theorem. Then, using variational methods together with truncation techniques, we prove a multiplicity theorem establishing the existence of at least five non-trivial solutions, with precise sign information for all of them (two positive solutions, two negative solutions and a nodal (sign changing) solution).

Keywords

scalar p-Laplaciancritical groupsmountain pass theoremC conditionp-superlinearityAR condition

MSC classification

Secondary: 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.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 805 - 827
Copyright
Copyright © Edinburgh Mathematical Society 2013 

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