In this paper, we prove the following general result. Let
X
X
be a real Hilbert space and
J
:
X
→
R
J:X\to \textbf {R}
a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that
\[
lim sup
‖
x
‖
→
+
∞
J
(
x
)
‖
x
‖
2
≤
0
.
\limsup _{\|x\|\to +\infty }{{J(x)}\over {\|x\|^2}}\leq 0\ .
\]
Then, for each
r
∈
]
inf
X
J
,
sup
X
J
[
r\in \ ]\inf _{X}J,\sup _{X}J[
for which the set
J
−
1
(
[
r
,
+
∞
[
)
J^{-1}([r,+\infty [)
is not convex and for each convex set
S
⊆
X
S\subseteq X
dense in
X
X
, there exist
x
0
∈
S
∩
J
−
1
(
]
−
∞
,
r
[
)
x_0\in S\cap J^{-1}(]-\infty ,r[)
and
λ
>
0
\lambda >0
such that the equation
\[
x
=
λ
J
′
(
x
)
+
x
0
x=\lambda J’(x)+x_0
\]
has at least three solutions.