From nonlinear Schrödinger equation to interacting particle system: 1 < p < 2

TL;DR

The paper analyzes the limiting arrangement of infinitely many peaks for the subcritical nonlinear Schrödinger equation as $\varepsilon\to 0$, deriving an interacting-particle system that governs the peak locations. It extends the range to $1<p<2$ by employing an integral-estimate approach on the error equation, in contrast to the p$\ge$2 case treated previously with reverse Lyapunov–Schmidt reduction, and connects the peak configuration to a weighted interaction law with $\ell_{\alpha\beta}$. The main contribution is the precise limiting condition $\sum_{\beta\neq\alpha} \ell_{\alpha\beta}\frac{\xi_{\beta}-\xi_{\alpha}}{|\xi_{\beta}-\xi_{\alpha}|}=0$ with $\ell_{\alpha\beta}\ge0$, $\sum_{\beta\neq\alpha}\ell_{\alpha\beta}=1$, and $\ell_{\alpha\beta}=0$ unless $\beta\in\mathcal{N}_{\alpha}$, where $\mathcal{N}_{\alpha}$ denotes the nearest neighbors. The analysis shows that, under mild separation and density hypotheses, the peaks become periodically distributed along the remaining direction, providing evidence for Wei's conjecture. The approach relies on sharp outer/inner error estimates, nondegeneracy of the ground state, and careful integral computations that capture the leading-order interactions among peaks.

Abstract

We investigate the limiting behavior of solutions with infinitely many peaks to nonlinear Schrödinger equations [-epsilon^2 Delta u_epsilon + u_epsilon = u_epsilon^p, u_epsilon > 0 in R^n,] as epsilon -> 0, where p is Sobolev subcritical. We derive the interaction law among the limiting peak points and complete the analysis for the previously unresolved range 1 < p < 2, extending the work of Ao, Lv, and Wang (J. Differential Equations, 2025).

Theorems & Definitions (19)

• Theorem 1.1
• Remark 1.1
• Lemma 2.1
• proof
• Lemma 2.2: Nondegeneracy
• Proposition 2.1: A priori estimate
• Proposition 2.2
• Remark 2.1
• Lemma 3.1
• proof