2025/07/22 by Leyun Wu, Chilin Zhang, Wu, Leyun +1
Computer Science · Mathematics · #35B40 #35J75 #45G05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Boundary (topology) #Cone (formal languages) #Differential Equations and Numerical Methods #FOS: Mathematics #Function (biology) #Growth rate #Ode #Recursion (computer science) #Representation (politics) #Sequence (biology) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2507.16319
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For γ>0, we study the sharp boundary growth rate estimate of solutions to the Dirichlet problem of the singular Lane-Emden-Fowler equation -Δu=u-γ in a critical C1,1 epigraphical cone ConeΣ. We show that the growth rate estimate exhibits fundamentally different behaviors in the following three cases: 1<γ<2, γ=2, and γ>2. Moreover, we obtain the sharp growth rate estimate near the origin for γ>1. As a consequence, we show that when ConeΣ is a C1,1 epigraphical cone, the additional solvability condition in \cite[Theorem 1.3]GuLiZh25 is both sufficient and necessary to achieve the growth rate therein, thereby resolving the main open question left in that paper. With the growth rate estimate, we also derive the optimal modulus of continuity for solutions via the interior Schauder estimate. Our approach is to control the values of a solution U(x) in the region Ω=ConeΣ∩ B1 by introducing a sequence of reference points pk=\frac161-k2en. From the Green function representation of U(x), we derive a discrete integral equation for the sequence ak=16kϕU(pk). Such a computation converts the original PDE problem into a recursion for a discrete integral equation, which can be effectively analyzed using basic ODE methods.