2025/09/11 by Haoyu Li, Cheng, Zetao, Lei Zhang +2
Computer Science · Engineering · Mathematics · #35J20 #35J47 #35J57 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2509.09781
openalex publication_date 2025/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the following Liouville system defined on a compact Riemann surface M, -Δui=∑j=1n aijρj(\frachj euj∫Ωhj euj-1) in M for i=1,⋯,n,where the coefficient matrix A=(aij)n× n is nonnegative, h1, …, hn are positive smooth functions, and ρ1, …, ρn are positive constants. For the blowup solutions, we establish their uniqueness and non-degeneracy based on natural assumptions. The main results significantly generalize corresponding results for single Liouville equations \citeBartJevLeeYang2019,BartYangZhang20241,BartYangZhang20242. To overcome several substantial difficulties, we develop certain tools and extend them into a more general framework applicable to similar situations. Notably, to address the considerable challenge of a continuum of standard bubbles, we refine the techniques from Huang-Zhang \citeHuangZhang2022 and Zhang \citeZhang2006,Zhang2009 to achieve extremely precise pointwise estimates. Additionally, to address the limited information provided by the Pohozaev identity, we develop a useful Fredholm theory to discern the exact role that the Pohozaev identity plays for systems. The considerable difference between systems and a single equation is also reflected in the location of blowup points, where the uncertainty of the energy type of the blowup point makes it difficult to determine the sufficiency of pointwise estimates. In this regard, we extend our highly precise pointwise estimates to any finite order. This aspect is drastically distinct from analyses of single equations.