  We study the existence of positive solutions of a particular elliptic system
in $\mathbb{R}^3$ composed of two coupled non linear stationary Schr\"odinger
equations (NLSEs), that is $-\epsilon^2 \Delta u + V(x) u= h_v(u,v), -
\epsilon^2 \Delta v + V(x) v=h_u (u,v)$. Under certain hypotheses on the
potential $V$ and the non linearity $h$, we manage to prove that there exists a
solution $(u_\epsilon,v_\epsilon)$ that decays exponentially with respect to
local minima points of the potential and whose energy tends to concentrate
around these points, as $\epsilon \to 0$. We also estimate this energy in terms
of particular ground state energies. This work follows closely what is done in
https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more
general non linearity and we restrict ourselves to the case where the domain is
$\mathbb{R}^3$.
