Normalized solutions for nonlinear Schr\"odinger systems

We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times H^1(\R^N)$ for systems of nonlinear Schr"odinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz one is led to coupled systems of elliptic equations of the form $$\{\begin{array}{rl}{\displaystyle -\text{De}{u}_1}& {\displaystyle ={\text{la}}_1{u}_1+f_1(u_1)+{\text{pa}}_{1}F(u_1,u_2),}\\ {\displaystyle -\text{De}{u}_2}& {\displaystyle ={\text{la}}_2{u}_2+f_2(u_2)+{\text{pa}}_{2}F(u_1,u_2),}\\ {\displaystyle u_1,u_2}& {\displaystyle \in H^1({\mathbb{R}}^N),N\ge 2,}\end{array}$$ and we are looking for solutions satisfying $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }{u}_1{\mathrm{\mid }}^2=a_1,{\int }_{{\mathbb{R}}^N}\mathrm{\mid }{u}_2{\mathrm{\mid }}^2=a_2$$ where $a_1>0$ and $a_2>0$ are prescribed. In the system ${\text{la}}_1$ and ${\text{la}}_2$ are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e.\ $f_i(u_i)={\mu }_i\mathrm{\mid }{u}_i{\mathrm{\mid }}^{p_i-1}{u}_i$, $F(u_1,u_2)=\text{be}\mathrm{\mid }{u}_1{\mathrm{\mid }}^{r_1}\mathrm{\mid }{u}_2{\mathrm{\mid }}^{r_2}$, with positive constants $\be, \mu_i, p_i, r_i$\mathrm{.}TheexponentsareSobolevsubcriticalbutmaybe$L^2$-supercritical:$p_1,p_2,r_1+r_2\in]2,2^*[,\setminus\left{2+\frac4N\right}$.