Abstract

We consider the problemwhere $\Omega \subset \mathbb{R}^3$ is a smooth and bounded domain, $\varepsilon,\,\gamma _1,\,\gamma _2>0,$ $v,\,V:\Omega \rightarrow \mathbb{R}$, $f:\mathbb{R}\rightarrow \mathbb{R}$. We prove that this system has a least-energy solution $v_\varepsilon $ which develops, as $\varepsilon \rightarrow 0^+$, a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schrödinger equation. Moreover the unique peak approaches the most curved part of $\partial \Omega $, i.e., where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem for the single-equation case. The technique employed is based on the so-called energy method, which consists in the derivation of an asymptotic expansion for the energy of the solutions in powers of $\varepsilon $ up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in $\Omega $