Multiplicity of critical orbits to nonlinear, strongly indefinite functionals with sign-changing nonlinear part

TL;DR

This paper develops an abstract critical point theorem for strongly indefinite functionals with sign-changing nonlinearities and applies it to Schrödinger and Maxwell equations, demonstrating the existence of infinitely many critical orbits.

Contribution

It introduces a new abstract critical point theorem applicable to sign-changing nonlinear problems and demonstrates its use in nonlinear Schrödinger and Maxwell equations.

Findings

Proved existence of infinitely many critical orbits for certain functionals.

Applied the theorem to Schrödinger equations with spectral gap conditions.

Extended results to Maxwell equations with singular potentials.

Abstract

We show an abstract critical point theorem about existence of infinitely many critical orbits to strongly indefinite functionals with sign-changing nonlinear part defined on a dislocation space with a discrete group action. We apply the abstract result to a Schr\"odinger equation $$ -\Delta u + V(x) u = f(u) - \lambda g(u) $$ with $0$ in the spectral gap of the Schr\"odinger operator $-\Delta + V(x)$, that appears in nonlinear optics, as well as to the equations with singular potentials arising from the study of cylindrically symmetric, electromagnetic waves to the system of Maxwell equations.