A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

Tobias Weth; E.N. Dancer; Juncheng Wei

doi:10.1016/j.anihpc.2010.01.009

JournalsaihpcVol. 27, No. 3pp. 953–969

A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

Tobias Weth
Mathematische Institut, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany
E.N. Dancer
School of Mathematics and Statistics, University of Sydney, Australia
Juncheng Wei
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

Download PDF

Abstract

We study the set of solutions of the nonlinear elliptic system

⎩ ⎨ ⎧ - Δ u + λ_{1} u = μ_{1} u^{3} + β v^{2} u - Δ v + λ_{2} v = μ_{2} v^{3} + β u^{2} v u, v > 0 in Ω, u = v = 0 in Ω, in Ω, on \partial Ω, (P)

in a smooth bounded domain $Ω \subset R^{N}$ , $N ⩽ 3$ , with coupling parameter $β \in R$ . This system arises in the study of Bose–Einstein double condensates. We show that the value $β = - μ_{1} μ_{2}$ is critical for the existence of a priori bounds for solutions of (P). More precisely, we show that for $β > - μ_{1} μ_{2}$ , solutions of (P) are a priori bounded. In contrast, when $λ_{1} = λ_{2}$ , $μ_{1} = μ_{2}$ , (P) admits an unbounded sequence of solutions if $β ⩽ - μ_{1} μ_{2}$ .

Cite this article

Tobias Weth, E.N. Dancer, Juncheng Wei, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, pp. 953–969

DOI 10.1016/J.ANIHPC.2010.01.009