黎曼流形上一类非线性薛定谔方程的曲线集中现象

Concentration Phenomena on Curves for Nonlinear Schro¨dinger Equations on Riemannian Manifolds

本文主要考虑如下非线性薛定湾方程ε2∆gu − V (y)u + Q(y)up = 0, u > 0, u ∈ H1(M)其中p > 1, ε > 0 足够小,(M, g) 是RN (N ≥ 3) 中二维无边界的紧致黎曼流形。 假设Γ 是相对于加权泛函∫ΓVσ Q1/2 −σ静止和非退化的闭合曲线,这里σ = p+1/p-1-1/2 ,在对函数V (y) 和Q(y) 加上某些限制的情况下,当ε → 0 且远离某些特定值时,方程的解存在且集中在曲线Γ 附近,并且在曲线的任何领域外是指数衰减的。

In this paper, we mainly consider the following nonlinear Schrodinger Equation ε2∆gu − V (y)u + Q(y)up = 0, u > 0, u ∈ H1(M), where p > 1, ε > 0 is a small parameter, (M, g) is a compact smooth 2-dimensional Riemannian manifold without boundary in RN (N ≥ 3) . Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arc length ∫ΓVσ Q1/2 −σ, where σ = p+1/p-1-1/2 . Under some constraints for V (y) and Q (y), we prove the existence of a solution uε concentrating along the whole of Γ, exponentially small in ε at any positive distance from it, provided that ε is small and away from certain critical numbers.