Variational Methods in the Study of Inequality Problems for Nonlinear Elliptic Systems with Lack of Compactness

Abstract

We establish the existence of an entire weak solution for a class of stationary Schr\"odinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof relies on Chang's version of the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework, a result of Rabinowitz \cite{rabi} related to entire solutions of the Schr\"odinger equation.