Concentrating standing waves for the fractional nonlinear Schr\"odinger equation
Concentrating standing waves for the fractional nonlinear Schr\"odinger equation
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DOI：
10.1016/j.jde.2013.10.006
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发表时间：
2013-07
期刊：
影响因子：
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通讯作者：
J. D'avila;M. Pino;Juncheng Wei
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文献类型：
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作者：
J. D'avila;M. Pino;Juncheng Wei
我们考虑半线性方程ε 2s （- Δ） su + V (x) u - u p= 0, u> 0, u∈h2 s （rn）其中0< s< 1,1 < p< N+ 2 s N - 2s, V (x)是一个足够光滑的势，具有inf R V (x)> 0, ε> 0是一个小数值。让wλ的径向基态(−Δ)s w w +λλλ−wλp = 0 H 2 s R (N),我们构建解决方案的形式uε(x)∼∑我k wλ= 1 ((x−ξ我ε)/ε),我在哪里λ= V(ξ我ε)和ξε方法合适的临界点通过减少Lyapunov-Schmidt变分诉,我们恢复各种结果存在已知的s = 1。特别地，这样的解存在于v的k个非退化临界点附近。对于s= 1，这对应于Floer和Weinstein[13]和Oh[24]，[25]的经典结果。
We consider the semilinear equation ε 2 s (− Δ) s u+ V (x) u− u p= 0, u> 0, u∈ H 2 s (R N) where 0< s< 1, 1< p< N+ 2 s N− 2 s, V (x) is a sufficiently smooth potential with inf R V (x)> 0, and ε> 0 is a small number. Letting w λ be the radial ground state of (− Δ) s w λ+ λ w λ− w λ p= 0 in H 2 s (R N), we build solutions of the form u ε (x)∼∑ i= 1 k w λ i ((x− ξ i ε)/ε), where λ i= V (ξ i ε) and the ξ i ε approach suitable critical points of V. Via a Lyapunov–Schmidt variational reduction, we recover various existence results already known for the case s= 1. In particular such a solution exists around k nondegenerate critical points of V. For s= 1 this corresponds to the classical results by Floer and Weinstein [13] and Oh [24],[25].