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Ordinary vectorp-Laplacian; non-smooth critical point theory; locally Lipschitz function; Clarke subdifferential; non-smooth Palais-Smale condition; homoclinic solution; problem at resonance; Poincaré-Wirtinger inequality; Landesman-Lazer type condition

In this paper we study second order non-linear periodic systems driven by the ordinary vectorp-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by thep-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman-Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.

Evgenia H Papageorgiou¹ Nikolaos S Papageorgiou¹

1. Department of Mathematics, National Technical University, Zografou Campus, Athens - 15780, Greece

Published

August 2004

Manuscript received

13 November 2002

Manuscript revised

1 October 2003