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Some new results on the SU(3) Toda system and Lin-Ni problem Yang, Wen
Abstract
In this thesis, we mainly consider two problems. First, we study the SU(3) Toda system. Let (M,g) be a compact Riemann surface with volume 1, h₁ and h₂ be a C¹ positive function on M and p1; p2 ∈ ℝ⁺. The SU(3) Toda system is the following one on the compact surface M [Formula and equation omitted] where ∆ is the Beltrami-Laplace operator, αq ≥ 0 for every q ∈ S₁, S₁ ⊂ M, Bq ≥ 0 for every q ∈ S₂,S₂ ⊂ M and q is the Dirac measure at q ∈ M. We initiate the program for computing the Leray-Schauder topological degree of SU(3) Toda system and succeed in obtaining the degree formula for p1 ∈ (0,4π)(4π,8π), p2 ∉ 4πℕ when S₁ = S₂ = 0. Second, we consider the following nonlinear elliptic Neumann problem {∆u-μu +uq =0 in Ω,u > 0 in Ω,au/av=0 on aΩ. where q=n+2/n-2, μ > 0 and Ω is a smooth and bounded domain in ℝn. Lin and Ni (1986) conjectured that for μ small, all solutions are constants. In the second part of this thesis, we will show that this conjecture is false for a general domain in n = 4, 6 by constructing a nonconstant solution.
Item Metadata
Title |
Some new results on the SU(3) Toda system and Lin-Ni problem
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2015
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Description |
In this thesis, we mainly consider two problems. First, we study the SU(3)
Toda system. Let (M,g) be a compact Riemann surface with volume 1, h₁
and h₂ be a C¹ positive function on M and p1; p2 ∈ ℝ⁺. The SU(3) Toda
system is the following one on the compact surface M
[Formula and equation omitted]
where ∆ is the Beltrami-Laplace operator, αq ≥ 0 for every q ∈ S₁, S₁ ⊂ M,
Bq ≥ 0 for every q ∈ S₂,S₂ ⊂ M and q is the Dirac measure at q ∈ M. We
initiate the program for computing the Leray-Schauder topological degree
of SU(3) Toda system and succeed in obtaining the degree formula for p1 ∈
(0,4π)(4π,8π), p2 ∉ 4πℕ when S₁ = S₂ = 0.
Second, we consider the following nonlinear elliptic Neumann problem
{∆u-μu +uq =0 in Ω,u > 0 in Ω,au/av=0 on aΩ. where q=n+2/n-2, μ > 0 and Ω is a smooth and bounded domain in ℝn. Lin
and Ni (1986) conjectured that for μ small, all solutions are constants. In
the second part of this thesis, we will show that this conjecture is false for
a general domain in n = 4, 6 by constructing a nonconstant solution.
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Genre | |
Type | |
Language |
eng
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Date Available |
2015-07-16
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivs 2.5 Canada
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DOI |
10.14288/1.0166383
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2015-09
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivs 2.5 Canada