Existence and bounds of positive solutions for a nonlinear Schroedinger system

Abstract

We prove that, for any real $\lambda$, the system $-\Delta u+\lambda u=u^3-\beta u{v}^2$, $-\Delta v+\lambda v=v^3-\beta v{u}^2$, $u,v\in H_0^1(\Omega ),$ 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.

16 pages