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Miscellaneous applications of certain minimax theorems. I (1510.05036v1)
Published 16 Oct 2015 in math.FA
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 $C1$ 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 $x\to {{1}\over {2}}|x|2-\lambda J(x)-\langle x,\tilde y\rangle$ .