2005/03/05 by Evgenia H Papageorgiou, Nikolaos S Papageorgiou
Mathematics · #math.AP #msc:34B15 #msc:34C25 #msc:34C37 #msc:34A60
published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 114, No. 3, August 2004, pp. 269-298 · 28 pages
arxiv created 2005/03/05 · arxiv updated 2009/12/01
In this paper we study second order non-linear periodic systems driven by the ordinary vector p-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 the p-Laplacian. In the last section of the paper we examine the scalar \hboxnon-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.