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

 $\begin{array}{cc}\left\{\begin{array}{c}-\Delta u+{\lambda }_{1}u={\mu }_{1}{u}^{3}+\beta {v}^{2}u\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\hfill \\ -\Delta v+{\lambda }_{2}v={\mu }_{2}{v}^{3}+\beta {u}^{2}v\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\hfill \\ u,v>0\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\phantom{\rule{2em}{0ex}}u=v=0\phantom{\rule{1em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega ,\hfill \end{array}& \text{(P)}\end{array}$
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 $\beta ⩽-\sqrt{{\mu }_{1}{\mu }_{2}}$.

@article{AIHPC_2010__27_3_953_0,
author = {Dancer, E.N. and Wei, Juncheng and Weth, Tobias},
title = {A priori bounds versus multiple existence of positive solutions for a nonlinear {Schr\"odinger} system},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {953--969},
publisher = {Elsevier},
volume = {27},
number = {3},
year = {2010},
doi = {10.1016/j.anihpc.2010.01.009},
zbl = {1191.35121},
mrnumber = {2629888},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2010.01.009/}
}
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Dancer, E.N.; Wei, Juncheng; Weth, Tobias. A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 953-969. doi : 10.1016/j.anihpc.2010.01.009. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.01.009/

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