Existence and orbital stability of standing waves for nonlinear Schr\"odinger systems

By Tian-Xiang Gou, Louis Jeanjean. 2015

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In this paper we investigate the existence of solutions in $H^1(R^N) \times H^1(R^N)$ for nonlinear Schr\"odinger systems of the form \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + r_1\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -\Delta u_2 &= \lambda_2 u_2 + \mu_2 |u_2|^{p_2 -2}u_2 + r_2 \beta |u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned} \right. \] under the constraints \[\int_{R^N}|u_1|^2 \, dx = a_1>0,\quad \int_{R^N}|u_2|^2 \, dx = a_2>0. \] Here $ N \geq 1, \beta >0, \mu_i >0, r_i >1, 2 <p_i < 2 + \frac{4}{N}$ for $i=1,2$ and $ r_1 + r_2 < 2 + \frac{4}{N}$. This problem is motivated by the search of standing waves for an evolution problem appearing in several physical models. Our solutions are obtained as constrained global minimizers of an associated functional. Note that in the system $\lambda_1$ and $\lambda_2$ are unknown and will correspond to the Lagrange multipliers. Our main result is the precompactness of the minimizing sequences, up to translation, and as a consequence we obtain the orbital stability of the standing waves associated to the set of minimizers.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a61d8f91c4a970d4062a4811378b64f6957a5fce

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In Control and Systems Engineering

  1. Houwang Li
  2. Wenming Zou
  3. Thomas Bartsch
  4. Nicola Soave
  5. Gongbao Li
  6. Hongyu Ye
  7. Louis Jeanjean

In Mathematical Physics

  1. Houwang Li
  2. Wenming Zou
  3. Thomas Bartsch
  4. Nicola Soave
  5. Louis Jeanjean
  6. Gianmaria Verzini
  7. R. Molle
  8. Xinfu Li

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