Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Abstract

In this paper we investigate the existence of solutions in $H^1(R^N)\times H^1(R^N)$ for nonlinear Schrödinger systems of the form \[ \left\{ \begin{aligned} -Δu_1 &= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + r_1β|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -Δu_2 &= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 + r_2 β|u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned} \right. \] 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 $N\ge 1,\beta >0,{\mu }_i>0,r_i>1,2<p_i<2+\frac{4}{N}$ 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.

References in corpus (1)

• A new rearrangement inequality and its application for L^2-constraint minimizing problems

Cited by in corpus (3)

• Existence and stability of standing waves for nonlinear Schrodinger systems involving the fractional Laplacian
• Normalized solutions to the Chern-Simons-Schrödinger system
• Normalized solutions for Schrödinger system with quadratic and cubic interactions