Existence and bounds of positive solutions for a nonlinear Schrödinger system

Noris, Benedetta; Ramos, Miguel

doi:10.1090/S0002-9939-10-10231-7

Existence and bounds of positive solutions for a nonlinear Schrödinger system
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by Benedetta Noris and Miguel Ramos PDF

Proc. Amer. Math. Soc. 138 (2010), 1681-1692 Request permission

Abstract:

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.

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