By Ambrosetti A., Colorado E., Ruiz D.
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Additional info for Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations
22] B. Sirakov, Least energy solitary waves for a system of nonlinear Schr¨ odinger equations in Rn , preprint.  M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birk¨auser Boston, MA, 1996.
Clearly, u, u and u satisfy u(x) + u (x) + u (x) ≤ Ce−bα1 |x| 41 for some C > 0. Therefore, we have that: uξ1 (x)uξ2 (x)uξ3 (x) dx ≤ C exp[−bα1 (|x − ξ1 | + |x − ξ2 | + |x − ξ3 |)] dx ≤ C exp[−b(α1 − α2 )|x − ξ1 |] exp[−bα2 (|x − ξ1 | + |x − ξ2 | + |x − ξ3 |)] dx ≤ C exp[−b(α1 − α2 )|x|] dx exp[−bα2 ξq0 ] ≤ C exp[−bα2 ξq0 ] ≤ C exp[−bqξ]. Acknowledgement. We thank Andrea Malchiodi for useful comments and discussions. C. R. thank SISSA for the warm hospitality during several stays in which this work has been accomplished.
Funct. 69 (1986), 397-408.  X. Kang, J. Wei, On interacting bumps of semi-classical states of nonlinear Schr¨ odinger equations, Adv. Differential Equations 5 (2000), 899-928. N. Lebedev, Special Functions and their Applications, Prentice Hall, 1965. Y. Li, On a singularly perturbed elliptic equation, Adv. Differential Equations 2 (1997), 955-980. , Ground state of N coupled nonlinear Schr¨ odinger equations in Rn , n ≤ 3, Comm. Math. Phys. 255 (2005), 629-653.  A. Malchiodi, J. M. Ni, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann.
Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations by Ambrosetti A., Colorado E., Ruiz D.