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 \(2r^{2}h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))\), \(h(r)>0\), 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 \(\mu _1\in \mathbb {R}\setminus \{0\}\).