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

Abstract

We study the set of solutions of the nonlinear elliptic system

in a smooth bounded domain $\Omega \subset {\mathbb{R}}^N$, $N⩽3$, with coupling parameter $\beta \in \mathbb{R}$. This system arises in the study of Bose–Einstein double condensates. We show that the value $\beta =-\sqrt{{\mu }_1{\mu }_2}$ is critical for the existence of a priori bounds for solutions of (P). More precisely, we show that for $\beta >-\sqrt{{\mu }_1{\mu }_2}$, solutions of (P) are a priori bounded. In contrast, when ${\lambda }_1={\lambda }_2$, ${\mu }_1={\mu }_2$, (P) admits an unbounded sequence of solutions if $\beta ⩽-\sqrt{{\mu }_1{\mu }_2}$.