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, p. 953-969
We study the set of solutions of the nonlinear elliptic system {-Δu+λ 1 u=μ 1 u 3 +βv 2 uinΩ,-Δv+λ 2 v=μ 2 v 3 +βu 2 vinΩ,u,v>0inΩ,u=v=0onΩ,(P) in a smooth bounded domain Ω N , N3, with coupling parameter β. 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 .
@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},
     publisher = {Elsevier},
     volume = {27},
     number = {3},
     year = {2010},
     pages = {953-969},
     doi = {10.1016/j.anihpc.2010.01.009},
     zbl = {1191.35121},
     mrnumber = {2629888},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_3_953_0}
}
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://www.numdam.org/item/AIHPC_2010__27_3_953_0/

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