We start by reviewing from a variational point of view the classical spectral theory of the Dirichlet-Laplacian. On a general open set, it is well-known that the spectrum may fail to be purely discrete. We then turn our attention to a nonlinear variant of this problem, by considering the case of the $p-$Laplacian with Dirichlet homogeneous conditions. More precisely, we analyze the minmax levels of the constrained $p-$Dirichlet integral: we show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincar\'e constant ``at infinity'', it actually defines an eigenvalue. We also prove a quantitative exponential fall-off at infinity for the relevant eigenfunctions: this can be seen as a generalization of classical \v{S}nol-Simon--type estimates to the nonlinear case.
Some of the results presented have been obtained in collaboration with Luca Briani (TUM Monaco), Giovanni Franzina (CNR-IAC) and Francesca Prinari (Pisa).