Miscellaneous applications of certain minimax theorems. I

TL;DR

This paper extends minimax theorems in Hilbert spaces to establish the existence of multiple solutions for certain nonlinear equations involving a $C^1$ functional with compact derivative, under specific growth conditions.

Contribution

It introduces new conditions under which multiple solutions exist for nonlinear equations in Hilbert spaces using minimax theorems, expanding previous theoretical frameworks.

Findings

Existence of at least three solutions for the nonlinear equation.

Two solutions are global minima of a related functional.

Results apply to dense convex subsets of the Hilbert space.

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 $x\to {{1}\over {2}}\|x\|^2-\lambda J(x)-\langle x,\tilde y\rangle$ .