Entire solutions of Schr\"{o}dinger elliptic systems with discontinuous nonlinearity and sign-changing potential

TL;DR

This paper proves the existence of entire solutions for a class of Schrödinger systems with discontinuous nonlinearities and sign-changing potentials, extending previous results to a nonsmooth setting using Clarke's critical point theory.

Contribution

It introduces a novel approach employing Clarke's critical point theory and Chang's Mountain Pass Lemma to handle discontinuous nonlinearities in Schrödinger systems.

Findings

Existence of entire solutions under subcritical discontinuous nonlinearities.

Generalization of Rabinowitz's result to nonsmooth potentials.

Application of Clarke's critical point theory to Schrödinger systems.

Abstract

We establish the existence of an entire solution for a class of stationary Schr\"{o}dinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply 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\"{o}dinger equation.