  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 $g=\frac{da^2}{h(a^2)}+a^2g_{S^n}$ for some function $h$
where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in
$\mathbb{R}^n$ for any $n\ge 2$. More precisely for any $\lambda\ge 0$ and
$c_0>0$, we prove that there exist infinitely many solutions $h\in
C^2((0,\infty);\mathbb{R}^+)$ for the equation
$2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$,
$h(r)>0$, in $(0,\infty)$ satisfying $\underset{\substack{r\to
0}}{\lim}\,r^{\sqrt{n}-1}h(r)=c_0$ 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.
