Multiple positive normalized solutions for nonlinear Schr\"odinger systems

We consider the existence of multiple positive solutions to the nonlinear Schr"odinger systems sets on $H^1({\mathbb{R}}^N)\times H^1({\mathbb{R}}^N)$, [ \left{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1

• \beta r_1 |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 + \beta r_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \end{aligned} \right. ] under the constraint $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }{u}_1{\mathrm{\mid }}^{2}dx=a_1,{\int }_{{\mathbb{R}}^N}\mathrm{\mid }{u}_2{\mathrm{\mid }}^{2}dx=a_2\mathrm{.}$$ Here $a_1,a_2>0$ are prescribed, ${\mu }_1,{\mu }_2,\beta >0$, and the frequencies ${\lambda }_1,{\lambda }_2$ are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when $N \geq 1, 2 < p_1, p_2 < 2 + \frac 4N, r_1, r_2 > 1, 2 + \frac 4N < r_1 + r_2 < 2^$,thesecondwhen$ N \geq 1, 2+ \frac 4N < p_1, p_2 < 2^, r_1, r_2 > 1, r_1 + r_2 < 2 + \frac 4N.$Inbothcases,assumingthat$\beta >0$ is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.