| Contents |
| We consider the problem of finding $(\lambda_1,\lambda_2,u_1,u_2)$ such that:
\[
\left\{
\begin{array}
\mbox{}\Delta u_1 +V_1(x)u_1 + \mu_1 u_1^3 +\beta u_1 u_2^2=\lambda_1 u_1,\\
\Delta u_2 +V_2(x)_2 + \mu_2 u_2^3 +\beta u_2 u_1^2=\lambda_2 u_2,\\
\int_\Omega u_1^2\, dx=\rho_1, \ \ \int_\Omega u_2^2\, dx=\rho_2,
\end{array}
\right.
\]
on the whole $\mathbb{R}^N$, $N=2,3$ (with trapping potentials), or in some bounded domain with Dirichlet boundary conditions. We study existence of solutions for different ranges of the parameters (focusing/defocusing, competitive/cooperative, weak/strong interaction, small/big mass). For some selected family of solutions we prove orbital stability. |
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