Radially symmetric solutions of a nonlinear singular elliptic equation

For any \(\lambda \ge 0\) $\lambda \ge 0$ , \(2\le n\le 4\) $2\le n\le 4$ and \(\mu _1\in \mathbb {R}\) ${\mu }_1\in \mathbb{R}$ , we will prove the existence of unique radially symmetric solution \(h\in C^2((0,\infty ))\cap C^1([0,\infty ))\) $h\in C^2((0,\infty ))\cap C^1([0,\infty ))$ for the nonlinear singular elliptic equation \(2r^{2}h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))\) $2r^{2}h(r)h_{\mathit{rr}}(r)=(n-1)h(r)(h(r)-1)+r{h}_r(r)(r{h}_r(r)-\lambda r-(n-1))$ , \(h(r)>0\) $h(r)>0$ , in \((0,\infty )\) $(0,\infty )$ satisfying \(h(0)=1\) $h(0)=1$ , \(h_r(0)=\mu _1\) $h_r(0)={\mu }_1$ . We also prove the existence of unique analytic solution of the about equation on \([0,\infty )\) $[0,\infty )$ for any \(\lambda \ge 0\) $\lambda \ge 0$ , \(n\ge 2\) $n\ge 2$ and \(\mu _1\in \mathbb {R}\) ${\mu }_1\in \mathbb{R}$ . Moreover we will prove the asymptotic behaviour of the solution h for any \(n\ge 2\) $n\ge 2$ , \(\lambda \ge 0\) $\lambda \ge 0$ and \(\mu _1\in \mathbb {R}\setminus \{0\}\) ${\mu }_1\in \mathbb{R}\backslash \{0\}$ .