摘要:本文主要考虑如下非线性薛定湾方程ε ²∆ _gu − V (y)u + Q(y)u ^p= 0, u > 0, u ∈ H ¹(M)其中p > 1, ε > 0 足够小，(M, g) 是R ^N(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∫ _ΓV ^σQ ^{1/2 −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。

Abstract:In this paper, we mainly consider the following nonlinear Schrodinger Equation ε ²∆ _gu − V (y)u + Q(y)u ^p= 0, u > 0, u ∈ H ¹(M), where p > 1, ε > 0 is a small parameter, (M, g) is a compact smooth 2-dimensional Riemannian manifold without boundary in R ^N(N ≥ 3) . Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arc length ∫ _ΓV ^σQ ^{1/2 −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 u _εconcentrating 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.