Two Applications of a Three Critical Points Theorem

Abstract

In this paper we prove the existence of two nontrivial solutions for an asymptotically linear beam equation (denoted by (B)) as well as the existence of two nontrivial solutions for a superquadratic noncooperative elliptic system (denoted from now on by (ES)). The proofs are based on a variational approach, where the associated functionals are strongly indefinite. That is, the spaces on which the functionals are defined naturally split as H=H1 H2 , with dim H1= dim H2=+ and the functionals are unbounded from below on H1 and from above on H2 . The existence of three distinct critical levels is proved by the coupling of two linking theorems. Of course, for such a purpose, sharp estimates on the geometrical linking conditions are needed and they are obtained by a careful analysis of the structure of the splittings. The abstract critical point result that will be employed is an extension to the strongly indefinite functionals of a recent theorem of Marino et al. (see [MMP2]). Indeed, in our situation the result of [MMP2] cannot be used since in their result requires that one of the two subspaces must be finite dimensional, an assumption resulting from the use of the finite dimensional topological degree in their proof. Furthermore, the natural extension of their result via the use of the Leray Schauder topological degree remains insufficient to treat the problem (B). On the other hand, we can prove a suitable abstract result (see Theorem 2.1) by exploiting the notion of limit relative category as defined in [FLRW], which allows for the infinite dimensional splitting without imposing special restriction on the form of the gradient of the functionals as comes from a Leray Schauder approach. article no. 0178