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A global approach to the Schrödinger-Poisson system: An Existence result in the case of infinitely many states

In this paper we prove the existence of a solution to a nonlinear Schr{ö}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.