Sign-changing multi-peak standing waves of the NLSE with a point interaction

Consider the following semilinear problem with a point interaction in ${\mathbb{R}}^N$: $$-{\mathrm{\Delta }}_{\alpha }u+\omega u=u\mathrm{\mid }u{\mathrm{\mid }}^{p-2},$$ where $N\in \{2,3\}$; $\omega >0$; $-{\mathrm{\Delta }}_{\alpha }$ denotes the Hamiltonian of point interaction with inverse $s$-wave scattering length $-(4\pi \alpha {)}^{-1}$ and we want to solve for $u\text{\hspace{0.17em}⁣}:{\mathbb{R}}^N\to \mathbb{R}$. By means of Lyapunov–Schmidt reduction, we prove that this problem has sign-changing multi-peak solutions when either (1) $N=2$, $\alpha \in \mathbb{R}$, $p_{\ast }<p\le 3$ and $\omega$ is sufficiently large or (2) $N=3$, $0<\alpha <\mathrm{\infty }$, $p_{\ast }<p<3$ and $\omega$ is sufficiently small, where $2.45<p_{\ast }:=\frac{9+\sqrt{113}}{8}<2.46$.