Ground states of the planar nonlinear Schr\"odinger--Newton system with a point interaction

TL;DR

This paper proves the existence of ground states for a nonlinear Schr"odinger--Newton system with point interaction in two dimensions, linking these states to standing wave solutions of the associated evolution equation.

Contribution

It establishes sufficient conditions for ground state existence and connects critical points of an energy functional to standing wave solutions in a novel setting.

Findings

Ground states exist under certain parameter conditions.

Critical points correspond to standing wave solutions.

The work extends understanding of Schr"odinger--Newton systems with point interactions.

Abstract

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schr\"odinger--Newton system with a point interaction: \[ \begin{cases} - \Delta_\alpha u = w u + \beta u |u|^{p - 2} &\text{on} ~ \mathbb{R}^2; \\ - \Delta w = 2 \pi |u|^2 &\text{on} ~ \mathbb{R}^2; \\ \|u\|_{L^2}^2 = c, \end{cases} \] where $p > 2$; $\alpha, \beta \in \mathbb{R}$ and $- \Delta_\alpha$ 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 \[ \mathrm{i} \psi' (t) = - \Delta_\alpha \psi (t) - (\log |\cdot| \ast |\psi (t)|^2) \psi (t) - \beta \psi (t) |\psi (t)|^{p - 2}. \]