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 $p>2$; $\alpha ,\beta \in \mathbb{R}$ 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 \[ \mathrm{i} ψ' (t) = - Δ_αψ(t) - (\log |\cdot| \ast |ψ(t)|^2) ψ(t) - βψ(t) |ψ(t)|^{p - 2}. \]

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