journal of differential equations 125, 521547 (1996) A Unified Approach to a Class of Strongly Indefinite Functionals David G. Costa* Department of Mathematical Sciences, University of Nevada at Las Vegas, Las Vegas, Nevada 89154; Departamento de Matematica, Universidade de Brasilia, 70910 Brasilia, DF Brazil and Celius A. Magalhaes* Departamento de Matematica, Universidade de Brasilia, 70910 Brasilia, DF Brazil Received August 9, 1994; revised January 4, 1995 1. Introduction Many problems concerning differential equations are of a variational nature and can be put in the form Lu ={ u F ( x, u ), ( P) where L : D( L)/H  H is an unbounded, selfadjoint operator on a closed subspace H of L 2 ( 0, R m ), 0/R N is a bounded domain and F: 0_R m  R is a Caratheodory function which is C 1 in the variables u # R m . In this case, with suitable growth conditions on F and for an appropriate Hilbert space E /  H, the weak solutions of ( P) are the criti- cal points of the functional I : E  R given by I ( u )=q( u )& | 0 F ( x, u ) dx, where q is the quadratic form on E corresponding to the operator L. Here we are interested in the situation that I is strongly indefinite in the sense that it is neither bounded from above or from below, even modulo subspaces of finite dimension or codimension. This is indeed the case for article no. 0039 521 0022-039696 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * Research partially supported by CNPqBrazil.