Non-linear second-order periodic systems with non-smooth potential

Details Non-linear second-order periodic systems with non-smooth potential Non-linear second-order periodic systems with non-smooth potential

2005-03-05

arxiv.org/abs/math/0503087v1

Proc. Indian Acad. Sci. (Math. Sci.), Vol. 114, No. 3, August 2004, pp. 269-298

Mathematics

Analysis of PDEs

28 pages

Scientific paper

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 \hbox{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.

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