Existence and bounds of positive solutions for a nonlinear Schrödinger system

We prove that, for any $\lambda\in\mathbb{R}$, the system $-\Delta u +\lambda u = u^3-\beta uv^2$, $-\Delta v+\lambda v =v^3-\beta vu^2$, $u,v\in H^1_0(\Omega),$ where $\Omega$ is a bounded smooth domain of $\mathbb{R}^3$, admits a bounded family of positive solutions $(u_{\beta}, v_{\beta})$ as $\beta \to +\infty$. An upper bound on the number of nodal sets of the weak limits of $u_{\beta}-v_{\beta}$ is also provided. Moreover, for any sufficiently large fixed value of $\beta >0$ the system admits infinitely many positive solutions.