2024/12/10 by Linlin Sun, Sun, Linlin, Xiaobao Zhu +1 · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2412.07537
Let (M, g) be a compact Riemann surface with area 1. We investigate the Toda system \begincases -Δu1 = 2ρ1(h1eu1-1) - ρ2(h2eu2-1),
-Δu2 = 2ρ2(h2eu2-1) - ρ1(h1eu1-1), \endcases on (M, g) where ρ1, ρ2 ∈ (0,4π], and h1 and h2 are two smooth functions on M.When some ρi equals 4π, the Toda system becomes critical with respect to the Moser-Trudinger inequality for it, making the existence problem significantly more challenging. In their seminal article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), Jost, Lin, and Wang established sufficient conditions for the existence of solutions the Toda system when ρ1=4π, ρ2 ∈ (0,4π) or ρ1=ρ2=4π, assuming that h1 and h2 are both positive. In our previous paper we extended these results to allow h1 and h2 to change signs in the case ρ1=4π, ρ2 ∈ (0,4π). In this paper we further extend the study to prove that Jost-Lin-Wang's sufficient conditions remain valid even when h1 and h2 can change signs and ρ1=ρ2=4π. Our proof relies on an improved version of the Moser-Trudinger inequality for the Toda system, along with edicated analyses similar to Brezis-Merle type and the use of Pohozaev identities.