Exact decay rate of a nonlinear elliptic equation related to the Yamabe\n flow

2012/11/14 by Shu-Yu Hsu, Hsu, Shu-Yu · 1 citation
Mathematics · #35B40 (Primary) 58J37 #35J70 #58J05 (Secondary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1211.3232

openalex publication_date 2012/11/14 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let 0<m<(n-2)/n, n>2, \α=(2\β +\ρ)/(1-m) and\n\β>m\ρ/(n-2-mn) for some constant \ρ>0. Suppose v is a radially\nsymmetric symmetric solution of \(n-1)/(m)\Δ v^m+\α v+\β\nx\⋅\∇ v=0, v>0, in Rⁿ. When m=(n-2)/(n+2), the metric\ng=v^4/(n+2)dx² corresponds to a locally conformally flat Yamabe shrinking\ngradient soliton with positive sectional curvature. We prove that the solution\nv of the above nonlinear elliptic equation has the exact decay rate\n\lim_r\→\∞r²v(r)^1-m=\(2(n-1)(n(1-m)-2))/((1-m)(\α\n(1-m)-2\β)).\n

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