AbstractIn this paper, we study the following nonlinear elliptic systems: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=\partial _{u_1}F(x,u)&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=\partial _{u_2}F(x,u)&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$
-
Δ
u
1
+
V
1
(
x
)
u
1
=
∂
u
1
F
(
x
,
u
)
x
∈
R
N
,
-
Δ
u
2
+
V
2
(
x
)
u
2
=
∂
u
2
F
(
x
,
u
)
x
∈
R
N
,
where $$u=(u_1,u_2):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^2$$
u
=
(
u
1
,
u
2
)
:
R
N
→
R
2
, F and $$V_i$$
V
i
are periodic in $$x_1,\ldots ,x_N$$
x
1
,
…
,
x
N
and $$0\notin \sigma (-\,\Delta +V_i)$$
0
∉
σ
(
-
Δ
+
V
i
)
for $$i=1,2$$
i
=
1
,
2
, where $$\sigma (-\,\Delta +V_i)$$
σ
(
-
Δ
+
V
i
)
stands for the spectrum of the Schrödinger operator $$-\,\Delta +V_i$$
-
Δ
+
V
i
. Under some suitable assumptions on F and $$V_i$$
V
i
, we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).