An alternative theorem for gradient systems

Abstract

In this paper, given two Banach spaces $X, Y$ and a $C^1$ functional $\Phi:X\times Y\to {\bf R}$, under general assumptions, we show that either $\Phi$ has a saddle-point in $X\times Y$ or, for each convex and dense set $S\subseteq Y$, there is some $\tilde y\in S$ such that $\Phi(\cdot,\tilde y)$ has at least three critical points in $X$, two of which are global minima. Also, an application to non-cooperative elliptic systems is presented.