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

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