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

Abstract

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.