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

Dancer, E.N.; Wei, Juncheng; Weth, Tobias

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 953-969.

We study the set of solutions of the nonlinear elliptic system

$$\begin{array}{cc}\{\begin{array}{c}-\Delta u+{\lambda }_{1}u={\mu }_1{u}^3+\beta v^{2}u\text{in}\Omega ,\hfill \\ -\Delta v+{\lambda }_{2}v={\mu }_2{v}^3+\beta u^{2}v\text{in}\Omega ,\hfill \\ u,v>0\text{in}\Omega ,u=v=0\text{on}\partial \Omega ,\hfill \end{array}& \text{(P)}\end{array}$$

in a smooth bounded domain $\Omega \subset {ℝ}^N$, $N⩽3$, with coupling parameter $\beta \in ℝ$. 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 $\beta ⩽-\sqrt{{\mu }_1{\mu }_2}$.

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