An alternative theorem for gradient systems

Abstract

In this paper, given two Banach spaces $X,Y$ and a $C^1$ functional $\Phi :X\times Y\to \mathbf{R}$, 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.

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Cited by in corpus (1)

• A more complete version of a minimax theorem