On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers

TL;DR

This work analyzes the ground states of the stationary nonlinear Schrödinger equation $-\Delta \phi + \phi = |\phi|^{2\sigma}\phi$ on $\mathbb{R}^d$ as the nonlinearity exponent $\sigma$ approaches the endpoints $0$ and $\sigma_* = 2/(d-2)$ (for $d\ge3$). It combines variational, spectral, and perturbative techniques to prove convergence of ground states $u_\sigma$ to the Gaussian Gausson $u_0$ as $\sigma\to0$, with an explicit first-order correction $\mu_0$ and rate bounds; it also shows convergence, after a suitable rescaling, to the Aubin–Talenti algebraic soliton as $\sigma\to\sigma_*$, with sharp asymptotics for the scaling parameter and, in high dimensions, convergence in $H^1_r$ and precise divergence of the peak $\alpha(\sigma)$. The analysis relies on a careful study of the linearized operators, implicit function arguments, and a detailed examination of the radial harmonic oscillator spectrum to ensure invertibility and control the remainder terms. These endpoint results clarify limitations of previous claims in low dimensions and provide a rigorous, quantitative bridge between the NLS ground states and the special Gaussian and algebraic soliton solutions, complemented by numerical verifications. Overall, the paper advances understanding of nonlinear endpoint behavior in radially symmetric NLS and informs stability and numerical approximation strategies near critical nonlinearities.

Abstract

We consider the ground states of the nonlinear Schr{ö}dinger equation, which stand for radially symmetric and exponentially decaying solutions on the full space. We investigate their behaviors at both endpoint powers of the nonlinearity, up to some rescaling to infer non-trivial limits. One case corresponds to the limit towards a Gaussian function called Gausson, which is the ground state of the stationary logarithmic Schr{ö}dinger equation. The other case, for dimension at least three, corresponds to the limit towards the Aubin-Talenti algebraic soliton. We prove strong convergence with explicit bounds for both cases, and provide detailed asymptotics. These theoretical results are illustrated with numerical approximations.

Figures (6)

• Figure 1: Maximum of the ground states $\|u\|_{L^{\infty}}$ versus $\sigma$ for $d=1,\ldots,5$ (see Remark \ref{['rem:stiffness']} for comments on the limit ${\sigma} \to {\sigma}_*$).
• Figure 2: Ground state profile $u_{\sigma}$ for $d=2$ in linear scale (left) and logarithmic scale (right), for different values of ${\sigma}$.
• Figure 3: Dependence of $\alpha(\sigma)/\alpha(0)$ versus $\sigma$ and the expected slopes \ref{['eq:slope_at_0']} for $d=3,4,5$.
• Figure 4: Ground state profile $w_{\sigma}$ for $d = 5$ in the linear scale (left) and the logarithmic scale (right).
• Figure 5: Dependence of $\underline{\epsilon}(\sigma)$ (solid curve) and the asymptotic approximation $\epsilon_0(\sigma)$ (dashed line) versus $\sigma \in \left[0.5,0.66\right]$ for $d = 5$.

Theorems & Definitions (65)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 3.1
• Lemma 4.1: Lemma 2.1 in WangZhang2019
• Proposition 4.2
• Remark 4.3
• proof
• Lemma 4.4
• proof