A generalised Nehari manifold method for a class of non linear Schrödinger systems in ${\mathbb{R}}^3$

We study the existence of positive solutions of a particular elliptic system in ${\mathbb{R}}^3$ composed of two coupled non linear stationary Schrödinger equations (NLSEs), that is $-{\epsilon }^2\mathrm{\Delta }u+V(x)u=h_v(u,v),-{\epsilon }^2\mathrm{\Delta }v+V(x)v=h_u(u,v)$. 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 ${\mathbb{R}}^3$.