Growth estimates on positive solutions of the equation Δu + K u^n + 2\over n - 2 = 0 in \Rⁿ

2000/02/01 by Man Chun Leung, Leung, Man Chun
Mathematics · #35J60 #53C21 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Meromorphic and Entire Functions #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35J60 #msc:53C21

paper · pdf · doi:10.48550/arxiv.math/0002005

15 pages

arxiv created 2000/02/01 · openalex publication_date 2000/02/01 · arxiv updated 2009/11/30 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

We construct unbounded positive C²-solutions of the equation Δu + K u^{(n + 2)/(n - 2)} = 0 in \Rⁿ (equipped with Euclidean metric g_o) such that K is bounded between two positive numbers in \Rⁿ, the conformal metric g = u^{4/(n - 2)} g_o is complete, and the volume growth of g can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on u, we obtain growth estimate on the L^{2n/(n - 2)}-norm of the solution and show that it has slow decay.

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