Sharp Stability of $Δu-u+|u|^{p-1}u$ near a finite sums of ground states

TL;DR

This work establishes sharp quantitative stability for the elliptic equation $Δu - u + |u|^{p-1}u$ near finite sums of ground states $Q$, with stability bounds that depend on the spatial dimension $d$ and nonlinearity power $p$. The authors derive dimension- and interaction-aware rates via a two-step decomposition $u=σ+ρ$ and precise interaction estimates between ground states, culminating in an $H^1$-norm bound of the form $ig\|u-\sum_{k=1}^{m} Q(\cdot+y_k)\big\|_{H^1} \le C F_{d,p}\big(\|Δu-u+|u|^{p-1}u\|_{H^{-1}}\big)$. They prove sharpness by constructing multi-bubble configurations that saturate the derived rates, and extend the framework to complex-valued settings with phase restrictions for the cubic case and to the single-ground-state regime with direct NLS applications. The results illuminate how ground-state interactions and dimensionality govern stability near multi-soliton configurations, with implications for variational inequalities, Struwe-type compactness, and soliton dynamics in nonlinear dispersive equations.

Abstract

We establish sharp quantitative stability estimates near finite sums of ground states. The results depend on the dimension and the order of nonlinearity.

Theorems & Definitions (20)

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• proof : Proof of Theorem \ref{['theorem1.2']}