Existence and bounds of positive solutions for a nonlinear Schrödinger system

Abstract

We prove that, for any γ ε ℝ, the system - Δu + γu = u 3 - βuv 2 , -Δv + γv = v 3 - βvu 2 , u,v ε -H 1 0 (Ω), where Ω is a bounded smooth domain of ℝ 3 , admits a bounded family of positive solutions (uβ ,vβ )as β → +∞.An upper bound on the number of nodal sets of the weak limits of uβ - vβ is also provided. Moreover, for any sufficiently large fixed value of β > 0 the system admits infinitely many positive solutions. © 2010 American Mathematical Society.