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=da²/h(a²)+a²gSⁿ 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 λ ≥ 0 and c₀>0 , we prove that there exist infinitely many solutions h∈ C²((0,∞ );R⁺) for the equation 2r²h(r)hᵣᵣ(r)=(n-1)h(r)(h(r)-1)+rhᵣ(r)(rhᵣ(r)-λ r-(n-1)) , $h(r)>0$ , in (0,∞ ) satisfying r→ 0lim \,r√n-1h(r)=c₀ and prove the higher-order asymptotic behavior 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 behavior near the origin.