A Multiplicity Theorem in R^(n)

The aim of this paper is to establish the following result: THEOREM 1. - Let $X$ be a finite-dimensional real Hilbert space, and let $J:X\to {\bf R}$ be a $C^1$ function such that $$\liminf_{\|x\|\to +\infty}{{J(x)}\over {\|x\|^2}}\geq 0.$$ Moreover, let $x_0\in X$ and $r, s\in {\bf R}$, with $0<r<s$, be such that $$\inf_{x\in X}J(x)<\inf_{\|x-x_0\|\leq s}J(x)\leq J(x_0)\leq \inf_{r\leq\|x-x_0\|\leq s}J(x).$$ Then, there exists $\hat\lambda> 0$ such that the equation $$x+\hat\lambda J'(x)=x_0$$ has at least three solutions. We will proceed as follows. We first give the proof of Theorem 1. Then, we discuss in detail the finite-dimensionality assumption on $X$. More precisely, we will show not only that it can not be dropped, but also that it is very hard to imagine some additional condition (different from being $x_0$ a local minimum of $J$) under which one could adapt the given proof to the infinite-dimensional case. We finally conclude presenting an application of Theorem 1 to a discrete boundary value problem