An alternative theorem for gradient systems

In this paper, given two Banach spaces $X,Y$ and a $C^1$ functional $\mathrm{\Phi }:X\times Y\to \mathbf{R}$, under general assumptions, we show that either $\mathrm{\Phi }$ 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 $\mathrm{\Phi }(\cdot ,\tilde{y})$ has at least three critical points in $X$, two of which are global minima. Also, an application to non-cooperative elliptic systems is presented.