Ground states of elliptic problems over cones

Abstract

Given a reflexive Banach space X , we consider a class of functionals ∈ C1(X , R) that donot behave in a uniform way, in the sense that the map t → (tu), t > 0, does not have auniform geometry with respect to u ∈ X . Assuming instead such a uniform behavior withinan open cone Y ⊂ X {0}, we show that has a ground state relative to Y . Some furtherconditions ensure that this relative ground state is the (absolute) ground state of . Severalapplications to elliptic equations and systems are given.