Ground states of elliptic problems over cones

Abstract

Given a reflexive Banach space X, we consider a class of functionals \(\Phi \in C^1(X,{\mathbb {R}})\) that do not behave in a uniform way, in the sense that the map \(t \mapsto \Phi (tu)\), \(t>0\), does not have a uniform geometry with respect to \(u\in X\). Assuming instead such a uniform behavior within an open cone \(Y \subset X \setminus \{0\}\), we show that \(\Phi \) has a ground state relative to Y. Some further conditions ensure that this relative ground state is the (absolute) ground state of \(\Phi \). Several applications to elliptic equations and systems are given.