We describe a result on the asymptotic behavior of the solutions of a system with two elliptic equations in
\mathbb{R}^{N}
R
N
involving a small parameter. More precisely, we study the system
\left\{ \begin{aligned} -\varepsilon^{2} \text{div}(a(x) \nabla u)+u & = Q_{u}(u,v)+\frac{\gamma}{2^*} K_u(u,v)\ \ \text{in } \mathbb{R}^N, \\ -\varepsilon^{2} \Delta v + b(x) v & = Q_{v}(u,v)+\frac{\gamma}{2^*} K_v(u,v)\ \ \text{in } \mathbb{R}^N, \\ u,v \in H^{1}(\mathbb{R}^N), &\ u(x),\ v(x)>0\ \ \text{for each } x \in\mathbb{R}^N, \end{aligned} \right.
{
−
ε
2
div
(
a
(
x
)
∇
u
)
+
u
=
Q
u
(
u
,
v
)
+
γ
2
∗
K
u
(
u
,
v
)
in
R
N
,
−
ε
2
Δ
v
+
b
(
x
)
v
=
Q
v
(
u
,
v
)
+
γ
2
∗
K
v
(
u
,
v
)
in
R
N
,
u
,
v
∈
H
1
(
R
N
)
,
u
(
x
)
,
v
(
x
)
>
0
for each
x
∈
R
N
,
where
2^*=2N/(N-2)
2
∗
=
2
N
/
(
N
−
2
)
,
N\geq 3
N
≥
3
,
\varepsilon>0
ε
>
0
,
a
a
and
b
b
are positive continuous potentials, and
Q
Q
and
K
K
are homogeneous functions with
K
K
having critical growth. We use the penalization method for system introduced by C. O. Alves [Local mountain pass for a class of elliptic system, J. Math. Analysis Appl. 335 (2007) 135–150] in order to find a family of solutions
(u_{\varepsilon}, v_{\varepsilon})
(
u
ε
,
v
ε
)
in
H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)
H
1
(
R
N
)
×
H
1
(
R
N
)
such that, if
\Pi_{\varepsilon,a}
Π
ε
,
a
and
\Pi_{\varepsilon, b}
Π
ε
,
b
are maximum points of
u_{\varepsilon}
u
ε
and
v_{\varepsilon}
v
ε
respectively, then
\lim_{\varepsilon \rightarrow 0^+}a(\Pi_{\varepsilon, a}) = \inf_{x \in \mathbb{R}^{N}} a(x) \ \ \ \text{and}\ \ \lim_{\varepsilon \rightarrow 0^+}b(\Pi_{\varepsilon, b})= \displaystyle\inf_{x \in \mathbb{R}^{N}} b(x).
lim
ε
→
0
+
a
(
Π
ε
,
a
)
=
inf
x
∈
R
N
a
(
x
)
and
lim
ε
→
0
+
b
(
Π
ε
,
b
)
=
inf
x
∈
R
N
b
(
x
)
.
Moreover, we relate the number of solutions with the topology of the set where the potentials
a
a
and
b
b
attain their minima. We consider the subcritical case
\gamma=0
γ
=
0
and the critical case
\gamma=1
γ
=
1
.