We start by reviewing 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, in this generality. 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 $L^p$ Poincaré 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 Snol-Simon--type estimates to the nonlinear case. Some of the results presented have been obtained in collaboration with Luca Briani (TUM Monaco) and Francesca Prinari (Pisa).
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006