Existence and bounds of positive solutions for a nonlinear Schroedinger system

We prove that, for any real $\lambda$, 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 $R^3$, 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 difference $u_{\beta}-v_{\beta}$ is also provided. Moreover, for any sufficiently large fixed value of $\beta >0$ the system admits infinitely many positive solutions.