Ground states of the defocusing NLSE with a point interaction

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.