An existence theorem for some semilinear elliptic systems

Abstract

An existence theorem is obtained for a class of semilinear, second order, uniformly elliptic systems obtained formally from a variational principle and modeled on nonlinear Helmholtz systems. Superlinear growth of the nonlinear term precludes application of standard methods to these systems. Indeed, we permit very rapid growth of the nonlinear term, so the underlying functional is not defined on the Hilbert space within which a solution is naturally sought. Mollification of the nonlinear term nonetheless results in the resulting functional satisfying the Palais–Smale condition; critical points are determined by solution of a dynamical system. The limit of vanishing mollification then produces a weak solution of the original problem.