Coupled nonlinear Schrödinger equations with point interaction: existence and asymptotic behaviour

In this paper we deal with the following weakly coupled nonlinear Schrödinger system \begin{align*} \begin{cases} - Δ_αu + ωu = |u|^2 u + βu |v|^2&\quad $\mathrm{i}\mathrm{n}$\ $\mathbb{R}$^2,\ - Δv + \tildeω v = |v|^2 v + β|u|^2 v&\quad $\mathrm{i}\mathrm{n}$\ $\mathbb{R}$^2, \end{cases} %\tag{${\mathcal{P}}_{\beta }$} \end{align*} where $-{\mathrm{\Delta }}_{\alpha }$ denotes the Laplacian operator with a point interaction, $\omega$ greater then a suitable positive constant, $\tilde{\omega }>0$, and $\beta \ge 0$. For any $\beta \ge 0$ this system admits the existence of a ground state solution which can have only one nontrivial component or two nontrivial components and which could be regular or singular. We analyse this phenomenon showing how this depends strongly on the parameters. Moreover we study the asymptotic behaviour of ground state solutions whenever $\beta \to \mathrm{\infty }$.