A generalised Nehari manifold method for a class of non-linear Schrödinger systems in ℝ³

We study the existence of positive solutions of a particular elliptic system in ℝ3 composed of two non linear stationary Schrödinger equations (NLSEs), that is -∈2Δu + V(x)u = hv(u, v), -∈2Δv + 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∈, v∈) that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as ∈ → 0. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in [6], although here we consider a more general non linearity and we restrict ourselves to the case where the domain is ℝ3.