From nonlinear Schr\"odinger equation to interacting particle system: 1 < p < 2

TL;DR

This paper studies the asymptotic behavior of solutions with multiple peaks to a nonlinear Schrödinger equation as a small parameter approaches zero, focusing on the range 1 < p < 2, and derives interaction laws among peaks.

Contribution

It extends the analysis of peak interactions in nonlinear Schrödinger equations to the previously unresolved subcritical range 1 < p < 2.

Findings

Derived the interaction law among peak points for 1 < p < 2

Extended previous work to new parameter range

Provided a complete analysis of multi-peak solutions

Abstract

We investigate the limiting behavior of solutions with infinitely many peaks to nonlinear Schr\"odinger 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).