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: $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{\alpha }u+\omega u+u\mathrm{\mid }u{\mathrm{\mid }}^{p-2}=0& \text{in }{\mathbb{R}}^N;\\ \mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{L}}^2}^2=\mu ,\end{array}$$ where $-{\mathrm{\Delta }}_{\alpha }$ 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 $\omega \in \mathbb{R}$, $u\text{\hspace{0.17em}⁣}:{\mathbb{R}}^N\to \mathbb{R}$. We remark that this kind of solutions does not exist in the framework of the defocusing NLSE without a point interaction.