Sign-changing multi-peak standing waves of the NLSE with a point interaction
Gustavo de Paula Ramos
TL;DR
This work investigates sign-changing, multi-peak standing waves for a nonlinear Schrödinger equation with a zero-range (point) interaction in dimensions two and three. By employing Lyapunov–Schmidt reduction around a carefully constructed multi-peak ansatz built from the ground state $\Phi$ and the Green’s function $G$, the authors reduce the problem to an auxiliary equation and a bifurcation (reduced) equation, establishing existence in two regimes: $N=2$ with large $\omega$ and $N=3$ with small $\omega$ (both with $p_*<p\le3$ or $p_*<p<3$ respectively). The analysis hinges on precise asymptotics and uniform invertibility of a linearized operator, along with sharp estimates for inter-peak interactions and the point-interaction contribution, culminating in a construction of $u_\eta$ that converges to a linear combination of translated ground states as $\eta\to\infty$. The results extend the theory of multi-peak NLSE solitons to models with point impurities, highlighting how zero-range interactions influence the existence and geometry of nodal patterns in semilinear regimes. The findings offer a new avenue for understanding complex, sign-changing soliton structures in singular perturbation settings with impurity modeling.
Abstract
Consider the following semilinear problem with a point interaction in $\mathbb{R}^N$: \[- Δ_αu + ωu = u |u|^{p - 2},\] where $N \in \{2, 3\}$; $ω> 0$; $- Δ_α$ denotes the Hamiltonian of point interaction with inverse $s$-wave scattering length $- (4 πα)^{- 1}$ 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) $N = 2$, $α\in \mathbb{R}$, $p_* < p \leq 3$ and $ω$ is sufficiently large or (2) $N = 3$, $0 < α< \infty$, $p_* < p < 3$ and $ω$ is sufficiently small, where $2.45 < p_* := \frac{9 + \sqrt{113}}{8} < 2.46$.