We study existence and multiplicity of solutions of the elliptic system
\[
{
−
Δ
u
=
H
u
(
x
,
u
,
v
)
a
m
p
;
in
Ω
,
−
Δ
v
=
−
H
v
(
x
,
u
,
v
)
a
m
p
;
in
Ω
,
u
(
x
)
=
v
(
x
)
=
0
on
∂
Ω
,
\begin {cases} -\Delta u =H_u(x,u,v) & \text {in $\Omega $}, \\ -\Delta v =-H_v(x,u,v) & \text {in $\Omega $}, \quad u(x) = v(x) = 0 \quad \text {on $\partial \Omega $}, \end {cases}
\]
where
Ω
⊂
R
N
,
N
≥
3
\Omega \subset \mathbb {R}^N, N\geq 3
, is a smooth bounded domain and
H
∈
C
1
(
Ω
¯
×
R
2
,
R
)
H\in \mathcal {C}^1(\bar {\Omega }\times \mathbb {R}^2, \mathbb {R})
. We assume that the nonlinear term
\[
H
(
x
,
u
,
v
)
∼
|
u
|
p
+
|
v
|
q
+
R
(
x
,
u
,
v
)
with
lim
|
(
u
,
v
)
|
→
∞
R
(
x
,
u
,
v
)
|
u
|
p
+
|
v
|
q
=
0
,
H(x,u,v)\sim |u|^p + |v|^q + R(x,u,v) \ \ \text {with} \ \ \lim _{|(u,v)|\to \infty }\frac {R(x,u,v)}{|u|^p+|v|^q}=0,
\]
where
p
∈
(
1
,
2
∗
)
p\in (1, \ 2^*)
,
2
∗
:=
2
N
/
(
N
−
2
)
2^*:=2N/(N-2)
, and
q
∈
(
1
,
∞
)
q\in (1, \ \infty )
. So some supercritical systems are included. Nontrivial solutions are obtained. When
H
(
x
,
u
,
v
)
H(x,u,v)
is even in
(
u
,
v
)
(u,v)
, we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if
p
>
2
p>2
(resp.
p
>
2
p>2
). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.