  We study here the behavior of the solutions to a $2\times 2$ semi-linear
cooperative system involving Schr\" odinger operators (considered in its
variational form): $$LU:=(-\Delta + q(x))U = AU+\mu U + F(x,U) \quad{\rm in}\
\mathbb{R}^N$$ $$U(x)_{|x|\rightarrow \infty} \rightarrow 0$$ where $q$ is a
continuous positive potential tending to $+\infty$ at infinity; $\mu$ is a real
parameter varying near the principal eigenvalue of the system; $U$ is a column
vector with components $u_1$ and $u_2$ and $A$ is a square cooperative matrix
with constant coefficient. $F$ is a column vector with components $f_1$ and
$f_2$ depending eventually on $U$.
