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

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.