Some new results on the SU(3) Toda system and Lin-Ni problem

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.Science, Faculty ofMathematics, Department ofGraduat