The aim of this paper is to establish the following result: THEOREM 1. - Let
X
X
be a finite-dimensional real Hilbert space, and let
J:X\to {\bf R}
J
:
X
→
R
be a
C^1
C
1
function such that
\liminf_{\|x\|\to +\infty}{{J(x)}\over {\|x\|^2}}\geq 0.
lim inf
∥
x
∥
→
+
∞
J
(
x
)
∥
x
∥
2
≥
0.
Moreover, let
x_0\in X
x
0
∈
X
and
r, s\in {\bf R}
r
,
s
∈
R
, with
0<r<s
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).
inf
x
∈
X
J
(
x
)
<
inf
∥
x
−
x
0
∥
≤
s
J
(
x
)
≤
J
(
x
0
)
≤
inf
r
≤
∥
x
−
x
0
∥
≤
s
J
(
x
)
.
Then, there exists
\hat\lambda> 0
λ
^
>
0
such that the equation
x+\hat\lambda J'(x)=x_0
x
+
λ
^
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
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
x
0
a local minimum of
J
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