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

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 $$\{\begin{array}{rl}{\displaystyle -\mathrm{\Delta }{u}_1}& {\displaystyle ={\lambda }_1{u}_1+{\mu }_1\mathrm{\mid }{u}_1{\mathrm{\mid }}^{p_1-2}{u}_1+r_1\beta \mathrm{\mid }{u}_1{\mathrm{\mid }}^{r_1-2}{u}_1\mathrm{\mid }{u}_2{\mathrm{\mid }}^{r_2},}\\ {\displaystyle -\mathrm{\Delta }{u}_2}& {\displaystyle ={\lambda }_2{u}_2+{\mu }_2\mathrm{\mid }{u}_2{\mathrm{\mid }}^{p_2-2}{u}_2+r_2\beta \mathrm{\mid }{u}_1{\mathrm{\mid }}^{r_1}\mathrm{\mid }{u}_2{\mathrm{\mid }}^{r_2-2}{u}_2,}\end{array}$$ under the constraints $${\int }_{R^N}\mathrm{\mid }{u}_1{\mathrm{\mid }}^{2}dx=a_1>0,{\int }_{R^N}\mathrm{\mid }{u}_2{\mathrm{\mid }}^{2}dx=a_2>0.$$ 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.