Ground states of the defocusing NLSE with a point interaction

TL;DR

This paper proves the existence of ground states for a specific class of defocusing nonlinear Schrödinger equations with point interactions in certain dimensions and parameter ranges, which do not exist without the point interaction.

Contribution

It establishes the existence of ground states in a normalized setting for the defocusing NLSE with point interactions, explicitly computing a threshold parameter.

Findings

Existence of ground states for N=2, α∈ℝ, p>2 when μ<μ₀

Existence of ground states for N=3, α<0, 2<p<3 when μ<μ₀

Ground states do not exist without point interactions in these cases.

Abstract

Suppose that either (i) $N = 2$, $\alpha \in \mathbb{R}$ and $p > 2$ or (ii) $N = 3$, $\alpha < 0$ and $2 < p < 3$. We prove that there exists an explicitly computable $\mu_0 = \mu_0 (N, \alpha, p) > 0$ such that if $0 < \mu < \mu_0$, then the following normalized semilinear elliptic problem with a point interaction admits ground states: \[ \begin{cases} - \Delta_\alpha u + \omega u + u |u|^{p - 2} = 0 &\text{in} ~ \mathbb{R}^N; \\ \|u\|_{\mathscr{L}^2}^2 = \mu, \end{cases} \] where $- \Delta_\alpha$ denotes the Laplacian of point interaction (centered at the origin) with inverse scattering length $- 2 (N - 1) \pi \alpha$ and we want to solve for $\omega \in \mathbb{R}$, $u \colon \mathbb{R}^N \to \mathbb{R}$. We remark that this kind of solutions does not exist in the framework of the defocusing NLSE without a point interaction.