Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations

We are concerned with the following nonlinear Schrödinger equation - ε 2 Δ u + V ( x ) u = | u | p - 2 u , u ∈ H 1 ( R N ) , where N ≥ 3 , 2 < p < 2 N N - 2 . For ε small enough and a class of V ( x ), we show the uniqueness of positive multi-bump solutions concentrating at k different critica...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2015-12, Vol.54 (4), p.4037-4063
Main Authors: Cao, Daomin, Li, Shuanglong, Luo, Peng
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
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Summary:We are concerned with the following nonlinear Schrödinger equation - ε 2 Δ u + V ( x ) u = | u | p - 2 u , u ∈ H 1 ( R N ) , where N ≥ 3 , 2 < p < 2 N N - 2 . For ε small enough and a class of V ( x ), we show the uniqueness of positive multi-bump solutions concentrating at k different critical points of V ( x ) under certain assumptions on asymptotic behavior of V ( x ) and its first derivatives near those points. Especially, the degeneracy of critical points is allowed in this paper.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-015-0930-2