In this article we study the existence of solutions for the elliptic system
\[
−
Δ
u
=
∂
H
∂
v
(
u
,
v
,
x
)
in
Ω
,
−
Δ
v
=
∂
H
∂
u
(
u
,
v
,
x
)
in
Ω
,
u
=
0
,
v
=
0
on
∂
Ω
.
\begin {array}{*{20}{c}} { - \Delta u = \frac {{\partial H}}{{\partial v}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ { - \Delta v = \frac {{\partial H}}{{\partial u}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ {u = 0,\quad v = 0\quad {\text {on}}\;\partial \Omega .} \\ \end {array}
\]
where
Ω
\Omega
is a bounded open subset of
R
N
{\mathbb {R}^N}
with smooth boundary
∂
Ω
\partial \Omega
, and the function
H
:
R
2
×
Ω
¯
→
R
H:{\mathbb {R}^2} \times \bar \Omega \to \mathbb {R}
, is of class
C
1
{C^1}
. We assume the function H has a superquadratic behavior that includes a Hamiltonian of the form
\[
H
(
u
,
v
)
=
|
u
|
α
+
|
v
|
β
where
1
−
2
N
>
1
α
+
1
β
>
1
with
α
>
1
,
β
>
1.
H(u,v) = |u{|^\alpha } + |v{|^\beta }\quad {\text {where}}\;1 - \frac {2}{N} > \frac {1}{\alpha } + \frac {1}{\beta } > 1\;{\text {with}}\;\alpha > 1,\beta > 1.
\]
We obtain existence of nontrivial solutions using a variational approach through a version of the Generalized Mountain Pass Theorem. Existence of positive solutions is also discussed.