Standing Waves for Nonlinear Schrodinger Equations with a General Nonlinearity: One and Two Dimensional Cases
- 31 May 2008
- journal article
- research article
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 33 (6) , 1113-1136
- https://doi.org/10.1080/03605300701518174
Abstract
For N = 1,2, we consider singularly perturbed elliptic equations ϵ2Δ u − V(x) u + f(u)= 0, u(x)> 0 on R N , lim|x|→∞ u(x)= 0. For small ϵ > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean ( 2007 Byeon , J. , Oshita , Y. ( 2004 ). Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations . Comm. PDE 29 : 1877 – 1904 . [Google Scholar] ).Keywords
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