Singular limit and exact decay rate of a nonlinear elliptic equation

For any n≥3n≥3, 00η>0, β>0β>0, α≤β(n−2)/mα≤β(n−2)/m, we prove the existence of radially symmetric solution of n−1mΔvm+αv+βx⋅∇v=0, v>0v>0, in RnRn, v(0)=ηv(0)=η, without using the phase plane method. When 00α=2β/(1−m)>0 and lim|x|→∞|x|α/βv(x)=Alim|x|→∞|x|α/βv(x)=A for some constant A>0A>0 if 2β/(1−m)>max(α,0)2β/(1−m)>max(α,0). For β>0β>0 or α=0α=0, we prove that the radially symmetric solution v(m)v(m) of the above elliptic equation converges uniformly on every compact subset of RnRn to the solution of an elliptic equation as m→0m→0.