Differential Systems with Strongly Indefinite Variational Structure

The Euler-Lagrange equations of Lagrangians with a strongly indefinite quadratic part are studied by means of the direct min-max method of Benci and Rabinowitz. The simplest case is a system of two coupled semilinear Poisson equations. Using an analytic framework of a suitable family of products of fractional Sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates.