Another proof of the existence of homothetic solitons of the inverse mean curvature flow

We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in R-n x R, n >= 2, of the form (r, y(r)) or (r(y), y), where r = vertical bar x vertical bar, x is an element of R-n, is the radially symmetric coordinate and y is an element of R. More precisely for any 1/n < lambda < 1/n-1 and mu < 0, we will give a new proof of the existence of a unique solution r(y) epsilon C-2( mu, infinity) n C([ mu, infinity)) of the equation r(yy)(y)/1+ r(y)(y)(2) = n - 1/r(y) - 1 + r(y)(y)(2)/lambda(r(y) - yr(y)(y)), r(y) > 0, in (mu, infinity) which satisfies r(mu) = 0 and r(y)(mu) = lim(y SE arrow) mu r(y)(y) = +infinity. We prove that there exist constants y(2) > y(1) > 0 such that r(y)(y) > 0 for any mu < y < y(1), (r)y(y(1)) = 0, r(y)(y) < 0 for any y > y(1), r(yy)(y) < 0 for any mu < y < y(2), r(yy)(y(2)) = 0 and r(yy)(y) > 0 for any y > y(2). Moreover, limy(->+infinity) r(y) = 0 and lim(y ->+infinity) yr(y)(y) = 0.