Radially symmetric solutions of a nonlinear singular elliptic equation

For any $\lambda\ge 0$, $2\le n\le 4$ and $\mu_1\in\mathbb{R}$, we will prove the existence of unique radially symmetric solution $h\in C^2((0,\infty))\cap C^1([0,\infty))$ for the nonlinear singular elliptic 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 $h(0)=1$, $h_r(0)=\mu_1$. We also prove the existence of unique analytic solution of the about equation on $[0,\infty)$ 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 $\mu_1\in\mathbb{R}\setminus\{0\}$. AMS 2020 Mathematics Subject Classification: Primary 35J70, 35J75 Secondary 53C21