We are concerned with the existence of multiple periodic solutions for differential equations involving Fisher-Kolmogorov perturbations of the relativistic operator of the form
−
[
ϕ
(
u
′
)
]
′
=
λ
u
(
1
−
|
u
|
q
)
,
\begin{equation*} -\left [\phi (u’)\right ]’=\lambda u(1-|u|^q), \end{equation*}
as well as for difference equations, of type
−
Δ
[
ϕ
(
Δ
u
(
n
−
1
)
)
]
=
λ
u
(
n
)
(
1
−
|
u
(
n
)
|
q
)
;
\begin{equation*} -\Delta \left [\phi (\Delta u(n-1))\right ]=\lambda u(n)(1-|u(n)|^q); \end{equation*}
here
q
>
0
q>0
is fixed,
Δ
\Delta
is the forward difference operator,
λ
>
0
\lambda >0
is a real parameter and
ϕ
(
y
)
=
y
1
−
y
2
(
y
∈
(
−
1
,
1
)
)
.
\begin{equation*} \displaystyle \phi (y)=\frac {y}{\sqrt {1- y^2}}\quad (y\in (-1,1)). \end{equation*}
The approach is variational and relies on critical point theory for convex, lower semicontinuous perturbations of
C
1
C^1
-functionals.