Ground states of elliptic problems over cones

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