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

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