Ground states of elliptic problems over cones

TL;DR

This paper establishes the existence of ground states for certain functionals over cones in Banach spaces, with applications to elliptic equations, under less restrictive uniformity conditions.

Contribution

It introduces a new approach to find ground states relative to cones in Banach spaces, relaxing uniformity assumptions and applying to elliptic problems.

Findings

Existence of ground states relative to cones in Banach spaces.

Conditions under which relative ground states are absolute ground states.

Applications to elliptic equations and systems.

Abstract

Given a reflexive Banach space $X$, we consider a class of functionals $\Phi \in C^1(X,\Re)$ 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.