A singular perturbation problem for a nonlinear Schrödinger system with three wave interaction

In this paper, we consider the locations of spikes of ground states for the following nonlinear Schrödinger system with three wave interaction

as \(\varepsilon \rightarrow +0\). In addition, we study the asymptotic behavior of a quantity \(\inf _{x \in {\mathbb {R}}^N} {\tilde{c}}({{\textbf{V}}}(x);\gamma )\) as \(\gamma \rightarrow \infty \) which determines locations of spikes. In particular, we give the sharp asymptotic behavior of a ground states of (\({{\mathcal {P}}}_\varepsilon \)) for \(\gamma \) sufficiently large and small, respectively. Furthermore, we consider when all the ground states of (\({{\mathcal {P}}}_\varepsilon \)) are scalar or vector.