2024/12/07 by LinLin Sun, Sun, LinLin, Xiaobao Zhu +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2412.05578
openalex publication_date 2024/12/07 · openalex created_date 2024/12/12 · openalex updated_date 2026/07/28
Let (M,g) be a compact Riemann surface with area 1, we shall study the Toda system \begincases -Δu1 = 2ρ1(h1eu1-1) - ρ2(h2eu2-1),
-Δu2 = 2ρ2(h2eu2-1) - ρ1(h1eu1-1), \endcases on (M,g) with ρ1=4π, ρ2∈(0,4π), h1 and h2 are two smooth functions on M. In Jost-Lin-Wang's celebrated article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), they obtained a sufficient condition for the existence of this Toda system when h1 and h2 are both positive. In this paper, we shall improve this result to the case h1 and h2 can change signs. We shall pursue a variational method and use the standard blowup analysis. Among other things, the main contribution in our proof is to show the blowup can only happen at one point where h1 is positive.