AbstractIn the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$
−
(
a
+
b
∑
k
∈
Z
|
Δ
u
k
−
1
|
2
)
Δ
2
u
k
−
1
+
V
k
u
k
=
f
k
(
u
k
)
k
∈
Z
,
where a, b are two positive constants and $V=\{V_{k}\}$
V
=
{
V
k
}
is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$
Δ
u
k
−
1
=
u
k
−
u
k
−
1
and $\Delta ^{2}=\Delta (\Delta )$
Δ
2
=
Δ
(
Δ
)
is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$
{
f
k
}
satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.