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ON THE MULTIPLICITY OF SOLUTIONS FOR NON-LINEAR PERIODIC PROBLEMS WITH THE NON-LINEARITY CROSSING SEVERAL EIGENVALUES

Published online by Cambridge University Press:  25 November 2009

SOPHIA TH. KYRITSI  and
NIKOLAOS S. PAPAGEORGIOU
SOPHIA TH. KYRITSI
Affiliation:
Department of Mathematics, Hellenic Naval Academy, Pireaus 18539, Greece e-mail: [email protected]
NIKOLAOS S. PAPAGEORGIOU
Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece e-mail: [email protected]
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Abstract

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In this paper we consider a non-linear periodic problem driven by the scalar p-Laplacian and with a non-smooth potential. We assume that the multi-valued right-hand-side non-linearity exhibits an asymmetric behaviour at ±∞ and crosses a finite number of eigenvalues as we move from −∞ to +∞. Using a variational approach based on the non-smooth critical-point theory, we show that the problem has at least two non-trivial solutions, one of which has constant sign. For the semi-linear (p = 2), smooth problem, using Morse theory, we show that the problem has at least three non-trivial solutions, again one with constant sign.

Keywords

34B1534C25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 271 - 302
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

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