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[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=u2q-2u has a unique positive radial solution Uλλ, which decays exponentially at infinity. Hence it is reasonable that the Schrödinger system -Δu1+u1=u12q-2u1-εb(x)u2qqu1q-2u1, -Δu2+u2=u22q-2u2-εb(x)u1qqu2q-2u2 has multiple-bump solutions which behave like Uλλ in the neighborhood of some points. For u=(u1, u2)∈H1(R3H1(R3), a nonlinear functional Iεε(u)=I1(u1)+I2(u2)-(ε)/(q)∫R3b(x)u1qqu2qqdx is defined, where I1(u1)=1/2‖u12-1/(2q)∫R3u12qdx and I2(u2)=1/2‖u22ωω-1/(2q)∫R3u22qdx. It is proved that the solutions of the system are the critical points of Iεε. Let Z be the smooth solution manifold of the unperturbed problem and TzzZ is the tangent space. The critical point of Iεε is rewritten as the form of z+w, where w∈(TzzZ). Using some properties of Iεε, it is proved that there exists a critical point of Iεε close to the form(∑ni=1U(x-ξii), ∑ni=1V(x-ξii))which is a multi-bump solution.

References:

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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]
Last Update: 2012-12-20