A generalised Nehari manifold method for a class of non linear Schr\"odinger systems in R³

We study the existence of positive solutions of a particular elliptic system in R³ composed of two coupled non linear stationary Schr\"odinger equations (NLSEs), that is -ε² Δ u + V(x) u= hᵥ(u,v), - ε² Δ v + V(x) v=hᵤ (u,v). Under certain hypotheses on the potential V and the non linearity h, we manage to prove that there exists a solution (u_ε,v_ε) that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as ε → 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 R³.