Miscellaneous applications of certain minimax theorems. I

Here is one of the results of this paper (with the convention ${{1}\over {0}}=+\infty$):Let$X$bearealHilbertspaceandlet$J:X\to {\bf R}$bea$C^1$functional,withcompactderivative,suchthat$$\alpha^:=\max\left {0,\limsup_{|x|\to +\infty}{{J(x)}\over {|x|^2}}\right }<\beta^:=\sup_{x\in X\setminus {0}}{{J(x)}\over {|x|^2}}<+\infty\ .Then, for every $\lambda\in \left ]{{1}\over {2\beta^*}}, {{1}\over {2\alpha^*}}\right [$ and for every convex set $C\subseteq X$ dense in $X$, there exists $\tilde y\in C$ such that the equationx=\lambda J’(x)+\tilde y$$ has at least three solutions, two of which are global minima of the functional $x\to \frac{1}{2}\mathrm{\parallel }x{\mathrm{\parallel }}^2-\lambda J(x)-⟨x,\tilde{y}⟩$ .