We present a new min–max approach to the search of multiple
T
T
–periodic solutions to a class of fourth order equations
\[
u
i
v
(
t
)
−
c
u
(
t
)
=
f
(
t
,
u
(
t
)
)
,
t
∈
[
0
,
T
]
,
u^{iv}(t)-c u(t)=f(t,u(t)),\hspace {5mm}t\in [0,T],
\]
where
f
(
t
,
u
)
f(t,u)
is continuous,
T
T
–periodic in
t
t
and satisfies a superlinearity assumption when
|
u
|
→
∞
|u|\to \infty
. For every
n
∈
N
n\in \mathbb {N}
, we prove the existence of a
T
T
–periodic solution having exactly
2
n
2n
zeroes in
(
0
,
T
]
(0,T]
.