Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric g=da(2)/h(a(2))+a(2)gS(n) for some function h where gSn is the standard metric on the unit sphere Sn in Rn for any n >= 2. More precisely for any lambda >= 0 and c0>0, we prove that there exist infinitely many solutions h is an element of C-2((0,infinity);R+) for the equation 2r(2)h(r)hrr(r)=(n-1)h(r)(h(r)-1)+rhr(r)(rhr(r)-lambda r-(n-1)), h(r)>0, in (0,infinity) satisfying limr -> 0rn root-1h(r)=c0 and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin.