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

Abstract

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger--Newton system with a point interaction: \[ \begin{cases} - Δ_αu = w u + βu |u|^{p - 2} &\text{on} ~ \mathbb{R}^2; \\ - Δw = 2 π|u|^2 &\text{on} ~ \mathbb{R}^2; \\ \|u\|_{L^2}^2 = c, \end{cases} \] where $p > 2$; $α, β\in \mathbb{R}$ and $- Δ_α$ denotes the Laplacian of point interaction with scattering length $(- 2 πα)^{- 1}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem \[ \mathrm{i} ψ' (t) = - Δ_αψ(t) - (\log |\cdot| \ast |ψ(t)|^2) ψ(t) - βψ(t) |ψ(t)|^{p - 2}. \]