Miscellaneous applications of certain minimax theorems. I

Abstract

Here is one of the results of this paper (with the convention ${{1}\over {0}}=+\infty$): Let $X$ be a real Hilbert space and let $J:X\to {\bf R}$ be a $C^1$ functional, with compact derivative, such that $$\alpha^*:=\max\left \{0,\limsup_{\|x\|\to +\infty}{{J(x)}\over {\|x\|^2}}\right \}<\beta^*:=\sup_{x\in X\setminus \{0\}}{{J(x)}\over {\|x\|^2}}<+\infty\ .$$ Then, for every $\lambda\in \left ]{{1}\over {2\beta^*}}, {{1}\over {2\alpha^*}}\right [$ and for every convex set $C\subseteq X$ dense in $X$, there exists $\tilde y\in C$ such that the equation $$x=\lambda J'(x)+\tilde y$$ has at least three solutions, two of which are global minima of the functional $x\to {{1}\over {2}}\|x\|^2-\lambda J(x)-\langle x,\tilde y\rangle$ .