Standing wave solutions of a quasilinear Schrödinger equation in the small frequency limit

This article is concerned with the quasilinear Schrödinger equation $$\mathrm{\Delta }u-\omega u+\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u+\delta \mathrm{\Delta }(\mathrm{\mid }u{\mathrm{\mid }}^2)u=0,$$ where $\delta >0$, $N=2$ and $p>1$ or $N\ge 3$ 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 $N\ge 3$) 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 $1<p\le 1+\frac{4}{N}$ and decreasing if $1+\frac{4}{N}<p\le \frac{N+2}{N-2}$. In the supercritical case, the monotonicity of $M(\omega )$ depends on the dimension, except in the regime $p\ge 3+\frac{4}{N}$, 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.