Ground states of the planar nonlinear Schrödinger–Newton system with a point interaction

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger–Newton system with a point interaction: \begin{equation*}\begin{cases}- \Delta_\alpha u + \omega u=w u+\beta u |u|^{p - 2}&\text{on} ~ \mathbb{R}^2;\\- \Delta w=2 \pi |u|^2&\text{on} ~ \mathbb{R}^2;\\\|u\|_{L^2}^2 = c,\end{cases}\end{equation*} where $p \gt 2$ ; $\alpha, \beta \in \mathbb{R}$ ; $c \gt 0$ and $- \Delta_\alpha$ denotes the Laplacian of point interaction with s-wave scattering length $(- 2 \pi \alpha)^{- 1}$ , the unknowns being $u \colon \mathbb{R}^2 \to \mathbb{C}$ , $w \colon \mathbb{R}^2 \to \lbrack0, \infty\lbrack$ and the Lagrange multiplier $\omega \in \mathbb{R}$ . Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem \begin{equation*}\mathrm{i} \psi' (t)=- \Delta_\alpha \psi (t)-(\log |\cdot| \ast |\psi (t)|^2)\psi (t)-\beta\psi (t)|\psi (t)|^{p - 2}.\end{equation*}