Singular limit and exact decay rate of a nonlinear elliptic equation

Abstract

For any n ≥ 3, 0 < m ≤ (n − 2)/n, and constants η > 0, β > 0, α, satisfying α ≤ β(n − 2)/m, we prove the existence of radially symmetric solution of n−1 m ∆v m + αv + βx · ∇v = 0, v > 0, in R n , v(0) = η, without using the phase plane method. When 0 < m < (n − 2)/n, n ≥ 3, and α = 2β/(1 − m) > 0, we prove that the radially symmetric solution v of the above elliptic equation satisfies lim |x|→∞ |x| 2 v(x) 1−m log |x|

Global behaviour of solutions of the fast diffusion equation

“…Theorem 2.1. (Theorem 1.3 of [Hs1] and its proof) Let v be the unique radially symmetric solution of (1.5) and w be given by (1.8). Then [DS] and Lemma 2.2 of [HK]…”

“…Let n ≥ 3, 0 < m < n−2 n , m n−2 n+2 , β > 0 and λ > 0. Let v λ,β (r) be the radially symmetric solution of (1.5) given by [Hs1]. Then there exists a constant K 0 independent of β and λ and a constant K(λ, β)…”

Yamabe flow: Steady solitons and Type II singularities

“…Moreover, all (R n , u 1−m β,λ δ i j ) are isometric to each other by conformal changes x → ax, a > 0. Hsu,in [H1] obtained the first order decay rate at infinity of a Yamabe steady soliton u β,λ . Namely, it was shown that…”

Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

“…where φ m (u) = u m /m if m 0 log u if m = 0 (1.2) and the associated elliptic equation [DKS], [Hs2], [Hs4], [Hu4], [Ar], [DK], [V3] and in the study of Ricci and Yamabe flow on manifolds [DS2], [H], [V2], [W]. When m > 1, it appears in modelling the evolution of various diffusion processes such as the flow of a gas through a porous medium [Ar].…”