Min-max levels transition in parametrized elliptic problems on unbounded domains

We discuss some attempts to apply the classical variational methods, in the spirit of Ambrosetti-Rabinowitz, to parametrized elliptic problems defined on unbounded domains. We construct some min-max classes and establish some estimates which allow, for instance, to conclude that, while for a generic coefficient there is always a solution for all small values of the parameter, in the case of a coefficient with a suitable exponential decay and in dimension N = 2 the set of the values of the parameter for which the problem has a nontrivial solution is unbounded.