Standing wave solutions of a quasilinear Schr\"odinger equation in the small frequency limit

TL;DR

This paper investigates the asymptotic behavior of standing wave solutions to a quasilinear Schrödinger equation as the frequency approaches zero, classifying regimes and analyzing the monotonicity of associated mass functions.

Contribution

It establishes the uniqueness, non-degeneracy, and detailed asymptotic analysis of solutions in different regimes, including the critical Lane-Emden-Fowler case, and studies the mass function's monotonicity.

Findings

Identification of three regimes ('subcritical', 'critical', 'supercritical') for the solutions.

Asymptotic descriptions of solutions as frequency tends to zero in each regime.

Monotonicity properties of the mass function M(ω) depending on p and dimension.

Abstract

This article is concerned with the quasilinear Schr\"odinger equation \[ \Delta u-\omega u+|u|^{p-1}u+\delta\Delta(|u|^2)u=0, \] where $\delta>0$, $N=2$ and $p>1$ or $N\ge3$ and $1<p<\frac{3N+2}{N-2}$. After proving uniqueness and non-degeneracy of the positive solution $u_\omega$ for all $\omega>0$, our main results establish the asymptotic behavior of $u_\omega$ in the limit $\omega\to 0^+$. Three different regimes arise, termed 'subcritical', 'critical' and 'supercritical', corresponding respectively (when $N\ge3$) to $1<p<\frac{N+2}{N-2}$, $p=\frac{N+2}{N-2}$ and $\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}$. In each case a limit equation is exhibited which governs, in a suitable scaling, the behavior of $u_\omega$ in the limit $\omega\to 0^+$. The critical case is the most challenging, technically speaking. In this case, the limit equation is the famous Lane-Emden-Fowler equation. A substantial part of our efforts is dedicated to the study of the function $\omega\mapsto M(\omega)=\int_{\mathbb{R}^N} u_\omega^2$. We find that, for small $\omega>0$, $M(\omega)$ is increasing if $1<p\le 1+\frac4N$ and decreasing if $1+\frac4N< p\le\frac{N+2}{N-2}$. In the supercritical case, the monotonicity of $M(\omega)$ depends on the dimension, except in the regime $p\ge 3+\frac4N$, where $M(\omega)$ is always decreasing close to $\omega=0$. The crucial role played by $M(\omega)$ for the orbital stability of the standing wave $e^{i\omega t}u_\omega$, and for the uniqueness of normalized ground states, is discussed in the introduction.