[1] Li Yuxiang, Wen Xuefei,. Existence of multi-bump solutions for coupled Schrödinger systems [J]. Journal of Southeast University (English Edition), 2012, 28 (4): 496-501. [doi:10.3969/j.issn.1003-7985.2012.04.022]

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Existence of multi-bump solutions for coupled Schrödinger systems()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

28

Issue:

2012 4

Page:

496-501

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2012-12-30

Info

Title:

Existence of multi-bump solutions for coupled Schrödinger systems

Author(s):

Li Yuxiang;  Wen Xuefei

Department of Mathematics, Southeast University, Nanjing 211189, China

Keywords:

coupled Schrö;  dinger system;  multi-bump solution;  variational reduction method

PACS:

O175.25

DOI:

10.3969/j.issn.1003-7985.2012.04.022

Abstract:

The Schrödinger equation -Δu+λ²u=u^2q-2u has a unique positive radial solution U_λλ, which decays exponentially at infinity. Hence it is reasonable that the Schrödinger system -Δu₁+u₁=u₁^2q-2u₁-εb(x)u₂^qqu₁^q-2u₁, -Δu₂+u₂=u₂^2q-2u₂-εb(x)u₁^qqu₂^q-2u₂ has multiple-bump solutions which behave like U_λλ in the neighborhood of some points. For u=(u₁, u₂)∈H¹(R³)×H¹(R³), a nonlinear functional I_εε(u)=I₁(u₁)+I₂(u₂)-(ε)/(q)∫_R³b(x)u₁^qqu₂^qqdx is defined, where I₁(u₁)=1/2‖u₁‖²-1/(2q)∫_R³u₁^2qdx and I₂(u₂)=1/2‖u₂‖²_ωω-1/(2q)∫_R³u₂^2qdx. It is proved that the solutions of the system are the critical points of I_εε. Let Z be the smooth solution manifold of the unperturbed problem and T_zzZ is the tangent space. The critical point of I_εε is rewritten as the form of z+w, where w∈(T_zzZ)^⊥. Using some properties of I_εε, it is proved that there exists a critical point of I_εε close to the form(∑ⁿ_i=1U(x-ξ_ii), ∑ⁿ_i=1V(x-ξ_ii))which is a multi-bump solution.

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Memo

Memo:

Biography: Li Yuxiang(1972—), male, doctor, associate professor, lieyx@seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.11171063), the Natural Science Foundation of Jiangsu Province(No.BK2010404).
Citation: Li Yuxiang, Wen Xuefei. Existence of multi-bump solutions for coupled Schrödinger systems[J].Journal of Southeast University(English Edition), 2012, 28(4):496-501.[doi:10.3969/j.issn.1003-7985.2012.04.022]

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