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: $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{\alpha }u=wu+\beta u\mathrm{\mid }u{\mathrm{\mid }}^{p-2}& \text{on }{\mathbb{R}}^2;\\ -\mathrm{\Delta }w=2\pi \mathrm{\mid }u{\mathrm{\mid }}^2& \text{on }{\mathbb{R}}^2;\\ \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2=c,\end{array}$$ where $p>2$; $\alpha ,\beta \in \mathbb{R}$ and $-{\mathrm{\Delta }}_{\alpha }$ denotes the Laplacian of point interaction with scattering length $(-2\pi \alpha {)}^{-1}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem $$\mathrm{i}{\psi }^{\mathrm{\prime }}(t)=-{\mathrm{\Delta }}_{\alpha }\psi (t)-(\log \mathrm{\mid }\cdot \mathrm{\mid }\ast \mathrm{\mid }\psi (t){\mathrm{\mid }}^2)\psi (t)-\beta \psi (t)\mathrm{\mid }\psi (t){\mathrm{\mid }}^{p-2}\mathrm{.}$$