A global approach to the Schr\"odinger-Poisson system: An Existence result in the case of infinitely many states

TL;DR

This paper establishes the existence of solutions for a nonlinear Schrödinger-Poisson eigenvalue problem in low dimensions using a novel global variational approach that characterizes the entire spectrum simultaneously.

Contribution

It introduces a new global variational method for determining the spectrum and eigenfunctions of the Schrödinger-Poisson system, applicable even with infinitely many states.

Findings

Proves existence of solutions in dimension less than 3.

Characterizes the entire spectrum and eigenfunctions at once.

Introduces a novel approach for spectral analysis of self-adjoint operators.

Abstract

In this paper we prove the existence of a solution to a nonlinear Schr{\"o}dinger--Poisson eigenvalue problem in dimension less than $3$. Our proof is based on a global approach to the determination of eigenvalues and eigenfunctions which allows us to characterize the complete sequence of eigenvalues and eigenfunctions at once, via a variational approach, and thus differs from the usual and less general proofs developed for similar problems in the literature. Our approach seems to be new for the determination of the spectrum and eigenfunctions for compact and self-adjoint operators, even in a finite dimensional setting.