Existence and bounds of positive solutions for a nonlinear Schrödinger system

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