Existence of self-similar solution of the inverse mean curvature flow

We will give a new proof of a recent result of P.~Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $mathbb{R}^n$, $nge 2$, of the form $u(x,t)=e^{lambda t}f(e^{-lambda t} x)$ for any constants $lambda>frac{1}{n-1}$ and $mu<0$ such that $f(0)=mu$. More precisely we will give a new proof of the existence of a unique radially symmetric solution $f$ of the equation $mbox{div},left(frac{ abla f}{sqrt{1+| abla f|^2}} right)=frac{1}{lambda}cdotfrac{sqrt{1+| abla f|^2}}{xcdot abla f-f}$ in $mathbb{R}^n$, $f(0)=mu$, for any $lambda>frac{1}{n-1}$ and $mu<0$, which satisfies $f_r(r)>0$, $f_{rr}(r)>0$ and $rf_r(r)>f(r)$ for all $r>0$. We will also prove that $lim_{rtoinfty}frac{rf_r(r)}{f(r)}=frac{lambda (n-1)}{lambda (n-1)-1}$.