Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - 2003 - Volume 46, Issue 01  

Research Article

ON THE EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS FOR EQUATIONS DRIVEN BY THE $p$-LAPLACIAN AND WITH A NON-SMOOTH POTENTIAL

Leszek Gasińskia1 and Nikolaos S. Papageorgioua2

a1 Jagiellonian University, Institute of Computer Science, ul. Nawojki 11, 30072 Cracow, Poland

a2 National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece (npapg@math.ntua.gr)

Abstract

In this paper we examine periodic problems driven by the scalar $p$-Laplacian. Using non-smooth critical-point theory and a recent multiplicity result based on local linking (the original smooth version is due to Brezis and Nirenberg), we prove three multiplicity results, the third for semilinear problems with resonance at zero. We also study a quasilinear periodic eigenvalue problem with the parameter near resonance. We prove the existence of three distinct solutions, extending in this way a semilinear and smooth result of Mawhin and Schmitt.

AMS 2000 Mathematics subject classification: Primary 34C25

(Received February 11 2002)

Keywords

  • non-smooth critical-point theory;
  • locally Lipschitz functional;
  • Clarke subdifferential;
  • non-smooth Palais–Smale condition;
  • local linking;
  • coercive functional