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 $1 0$, 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 $1 0$, $M(\omega)$ 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.