2011/07/14 by Hsu, Shu-Yu · 2 citations
#35B40 Secondary 34C11 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 35J60
paper · doi:10.48550/arxiv.1107.2735
For any n≥ 3, 00, β>0, α, satisfying α≤β(n-2)/m, we prove the existence of radially symmetric solution of (n-1)/(m)Δvm+αv +βx⋅∇ v=0, v>0, in \Rn, v(0)=η, without using the phase plane method. When 00, we prove that the radially symmetric solution v of the above elliptic equation satisfies lim|x|→∞\frac|x|2v(x)1-mlog |x| =(2(n-1)(n-2-nm))/(β(1-m)). In particular when m=(n-2)/(n+2), n≥ 3, and α=2β/(1-m)>0, the metric gij=v(4)/(n+2)dx2 is the steady soliton solution of the Yamabe flow on \Rn and we obtain lim|x|→∞\frac|x|2v(x)1-mlog |x|=\frac(n-1)(n-2)β. When 0max (α,0), we prove that lim|x|→∞|x|α/βv(x)=A for some constant A>0. For β>0 or α=0, we prove that the radially symmetric solution v(m) of the above elliptic elliptic equation converges uniformly on every compact subset of \Rn to the solution u of the equation (n-1)Δlog u+αu+βx⋅∇ u=0, u>0, in \Rn, u(0)=η, as m→ 0.