2022/02/28 by Seunghyeok Kim, Kim, Seunghyeok, Sang-Hyuck Moon +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2202.13599
openalex publication_date 2022/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We concern a family \(uε,vε)\ε > 0 of solutions of the Lane-Emden system on a smooth bounded convex domain Ω in ℝN \begincases -Δuε = vεp amp;in Ω,
-Δvε = uε^qε amp;in Ω,
uε, vε gt; 0 amp;in Ω,
uε = vε =0 amp;on ∂Ω\endcases for N ≥ 4, max\1,(3)/(N-2)\ < p < qε and small ε := (N)/(p+1) + \fracNqε+1 - (N-2) gt; 0. This system appears as the extremal equation of the Sobolev embedding W2,(p+1)/p(Ω) \hookrightarrow L^qε+1(Ω), and is also closely related to the Calderón-Zygmund estimate. Under the a natural energy condition supε gt; 0 (‖uε‖_W^2,p+1 \over p(Ω) + ‖vε‖_W^2,qε+1 \over qε(Ω)) lt; ∞, we prove that the multiple bubbling phenomena may arise for the family \(uε,vε)\ε > 0, and establish a detailed qualitative and quantitative description. If p < (N)/(N-2), the nonlinear structure of the system makes the interaction between bubbles so strong, so the determination process of the blow-up rates and locations is completely different from that of the classical Lane-Emden equation. If p ≥ (N)/(N-2), the blow-up scenario is relatively close to (but not the same as) that of the classical Lane-Emden equation, and only one-bubble solutions can exist. Even in the latter case, the standard approach does not work well, which forces us to devise a new method. Using our analysis, we also deduce a general existence theorem valid on any smooth bounded domains.