A new proof for the existence of rotationally symmetric gradient Ricci solitons

We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension $n+1$, $2\le n\le 4$, with metric $g=\frac{d{a}^2}{h(a^2)}+a^{2}d\sigma$ for some function $h$ where $d\sigma$ is the standard metric on the unit sphere $S^n$ in ${\mathbb{R}}^n$. More precisely for any $\lambda \ge 0$, $2\le n\le 4$ and ${\mu }_1\in \mathbb{R}$, we prove the existence of unique solution $h\in C^2((0,\infty))\cap C^1([0,\infty))$fortheequation$2r^2h®h_{rr}®=(n-1)h®(h®-1)+rh_r®(rh_r®-\lambda r-(n-1))$,$h®>0$,in$(0,\infty)$satisfying$h(0)=1$,$h_r(0)=\mu_1$. We also prove the existence of unique analytic solution of the about equation on $[0,\mathrm{\infty })$ for any $\lambda \ge 0$, $n\ge 2$ and ${\mu }_1\in \mathbb{R}$. Moreover we will prove the asymptotic behaviour of the solution $h$ for any $n\ge 2$, $\lambda \ge 0$ and ${\mu }_1\in \mathbb{R}\setminus \{0\}$.