Instability of standing waves for a class of inhomogeneous Schrödinger equations with harmonic potential

Abstract

This notes studies the inhomogeneous non-linear Schrödinger equations with a harmonic potential

$$\begin{aligned} i\partial _tu +\Delta u-|x|^2u+|x|^{b}|u|^{p-1}u=0. \end{aligned}$$

Indeed, following the methods of Fukuizumi and Ohta (Differ Integral Equ 16(6):691–706, 2003), Ohta (Funccialaj Ekvacioj 61:135–143, 2018), the non-linear and strong instability of standing waves are obtained in the two different regimes $b>0$ and $b<0$.

Appendix: Proof of Lemma 2.11

Take $\epsilon >0$.

• A. First case $-2<b<0$.

  Take $(u_n)$ a bounded sequence of H. Without loss of generality, we assume that $(u_n)$ converges weakly to zero in H. Our purpose is to prove that $\Vert u_n\Vert _{L^{p}(|x|^{b}\,dx)}\rightarrow 0.$ Using Hölder estimate, if $(q,q')$ satisfying $q'|b|<N$, one gets

  $$\begin{aligned} \int _{|x|\le \epsilon }|x|^{b}|u_n|^{p}\,dx\le & {} \Vert u_n\Vert _{pq}^{p}\Vert |x|^{b}\Vert _{q'}\\\le & {} C\Vert u_n\Vert _{pq}^{p}\int _0^\epsilon \frac{dt}{t^{-q'b-N+1}}\\\le & {} C\Vert u_n\Vert _{pq}^{p}\epsilon ^{N+q'b}. \end{aligned}$$

  • 1. First sub-case $N\ge 3$.

    Now, since $p<p^*$, we have $N+\frac{2Nb}{2N-p(N-2)}>0$. Taking $q:=\frac{2N}{p(N-2)}$ and using Sobolev injections, it follows that there exists $\mu >0$ such that

    $$\begin{aligned} \int _{|x|\le \epsilon }|x|^{b}|u_n|^{p}\,dx\le & {} C\Vert u_n\Vert _{H^1}^{p}\epsilon ^{N+q'b}\\\le & {} C\epsilon ^{N+\frac{2Nb}{2N-p(N-2)}}\\\le & {} C\epsilon ^\mu . \end{aligned}$$

  • 2. Second sub-case $N=2$.

    Taking $q>>1$ and using Sobolev injections, it follows that for $b>-N$, there exists $\mu >0$ such that

    $$\begin{aligned} \int _{|x|\le \epsilon }|x|^{b}|u_n|^{p}\,dx\le & {} C\Vert u_n\Vert _{H^1}^{p}\epsilon ^{N+q'b}\\\le & {} C\epsilon ^{N+q'b}\\\le & {} C\epsilon ^\mu . \end{aligned}$$

  On the other hand, by Sobolev injections,

  $$\begin{aligned} \int _{|x|\ge \epsilon }|x|^{b}|u_n|^{p}\,dx \le \epsilon ^{b}\Vert u_n\Vert _{p}^{p}. \end{aligned}$$

  Since $\epsilon$ is arbitrary, one obtains $\int _{\mathbb {R}^N}|x|^{b}|u_n|^{p}\,dx\rightarrow 0$ as $n\rightarrow \infty .$ The proof is complete.

• B. Second case $b\ge 0$.

  Take $(u_n)$ a bounded sequence of $\Sigma .$ Without loss of generality, we assume that $(u_n)$ converges weakly to zero in $\Sigma .$ Our purpose is to prove that $\Vert u_n\Vert _{L^{p}(|x|^{b}\,dx)}\rightarrow 0.$ Write

  $$\begin{aligned} \int _{|x|\le \epsilon }|x|^{b}|u_n|^{p}\,dx\le C\Vert u_n\Vert _{p}^{p}\rightarrow 0. \end{aligned}$$

  Moreover, with Strauss inequality and Rellich Theorem,

  $$\begin{aligned} \int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|x|^{b}|u_n|^{p}\,dx\le C\Vert u_n\Vert _{L^\infty (\epsilon \le |x|\le \frac{1}{\epsilon })}^{p-2} \int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|u_n|^2\,dx\rightarrow 0. \end{aligned}$$

  Now, with Strauss inequality

  $$\begin{aligned} \int _{|x|\ge \frac{1}{\epsilon }}|x|^{b}|u_n|^{p}\,dx= & {} \int _{|x|\ge \frac{1}{\epsilon }}|x|^{b-2-(p-2)\frac{N-1}{2}}(|x|^{\frac{N-1}{2}}|u_n|)^{p-2}|x|^2|u_n|^2\,dx\\\le & {} C\int _{|x|\ge \frac{1}{\epsilon }}|x|^{b-2-(p-2)\frac{N-1}{2}}|x|^2|u_n|^2\,dx\\\le & {} C\epsilon ^{(p-2)\frac{N-1}{2}-b+2}. \end{aligned}$$

  The proof is achieved because ${b-2-(p-2)\frac{N-1}{2}}<0.$