2023/10/20 by Jungang Li, Guozhen Lu, Li, Jungang +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2310.13811
openalex publication_date 2023/10/20 · openalex created_date 2023/10/25 · openalex updated_date 2026/07/28
The classification of solutions to semilinear partial differential equations, as well as the classification of critical points of the corresponding functionals, have wide applications in the study of partial differential equations and differential geometry. The classical moving plane method and the method of moving sphere on the Euclidean space ℝn provide an effective approach to capture the symmetry of solutions. As far as we know, the moving sphere method has yet to be developed on the hyperbolic space ℍn. In the present paper, we focus on the following equation Pk u = f(u) on hyperbolic spaces ℍn, where Pk denotes the GJMS operators on ℍn and f : ℝ → ℝ satisfies certain growth conditions. We develop a moving sphere approach on ℍn to obtain the symmetry propertyas well as the classification of positive solutions to the above equation. Our methods also rely on the Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic space together with a Kelvin transform we introduce on the hyperbolic space in this paper. We also present applications to the higher order prescribed Q-curvature problem on the hyperbolic space.