Diierential Systems with Strongly Indeenite Variational Structure

We study the existence of non-trivial solutions for a class of problems containing in particular the following system of two coupled semilinear Poisson equations: (I) 8 > < > : ? v = f(u) in ; ? u = g(v) in ; u = v = 0 on @: (1) (2) (3) Here is a bounded domain in R N with a smooth boundary, and is the Laplace operator. Problem (I) allows for a variational formulation, i.e. solutions arise as critical points of the Lagrangian (4) where the functions F and G are the primitives of f and g. The objective of this paper is to establish a natural functional analytic frame work for the study of L, and to obtain existence results for critical points of L by means of the topological min-max approach due to Benci and Rabinowitz BR]. The quadratic part of L, A(u) = Z rurv = (u; v) H 1 0 ; is strongly indeenite. Indeed, if we replace (u; v) by (u; ?v), this results in a sign change of A, so that u = (u; v) = (0; 0) is a saddle point for A having an innnite Morse index: any decomposition of the u-space into two subspaces H 1 and H 2 such that A restricted to H 1 has a minimum in (0; 0), and A restricted to H 2 has a maximum in (0; 0), will necessarily imply that both H 1 are innnite dimensional. Hence neither the mountain pass theorem AR], where one usually has a local minimum in the origin, nor the saddle point theorem R1], which requires one of the two subspaces H 1 and H 2 to be nite dimensional, are of use here. Strongly indeenite functionals in the context of Hamiltonian systems and semilinear wave equations were studied in A,R3]. A direct min-max method for these functionals was introduced by Benci and Rabinowitz in BR], see also R2,Ho]. Their result, which we shall refer to in this paper as the "Indeenite Functional Theorem", is an extension of both the mountain pass theorem and the saddle point theorem to functionals, deened on a Hilbert space H, of the form L(u) = 1 2 (Lu; u) H ? H(u): (5) Here L is a bounded linear selfadjoint operator on H, and H : H ! H is nonlinear, satisfying a number of assumptions. This theorem allows a …