2020/03/07 by João Henrique Andrade, Andrade, João Henrique, João Marcos Ó +2 · 1 citation
Mathematics · #35B09 #35B40 #35J30 #35J60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35B09 #msc:35B40 #msc:35J30 #msc:35J60
paper · pdf · doi:10.48550/arxiv.2003.03487
53 pages
openalex publication_date 2020/03/07 · arxiv created 2021/02/25 · arxiv updated 2021/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study the asymptotic behavior for singular solutions to a critical fourth order system generalizing the constant Q-curvature equation. Our main result extends to the case of strongly coupled systems, the celebrated asymptotic classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1989)] and [N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Invent. Math., (1999)]. On the technical level, we use an involved spectral analysis to study the Jacobi fields' growth properties in the kernel of the linearization of our system around a blow-up limit solution. Besides, we obtain sharp a priori estimates for the decay rate of singular solutions near the origin. Consequently, we prove that sufficiently close to the isolated singularity solutions behave like the so-called Emden--Fowler solution. Our main theorem positively answers a question posed by [R. L. Frank and T. König, Anal. PDE (2019)] concerning the local behavior close to the isolated singularity for scalar solutions in the punctured ball.