Positive solutions for a singular second-order ordinary differential equation (Q875226)

The authors consider the positive solutions for the singular second-order ordinary differential equation \[ \frac{\varphi''}{\sqrt{1+| \varphi'| ^2}}-\lambda\frac{| \varphi'| ^2}{\varphi(1+| \varphi'| ^2)^m}+f(t)=0,\quad 0<t<1, \] with one of the following boundary conditions \[ \varphi(1)=\varphi(0)=0, \] \[ \varphi(1)=\varphi(0)=\varphi'(1)=\varphi'(0)=0, \] where \(\lambda>0, m\geq\frac{1}{2}, f(t)\in C^1[0,1]\) and \(f(t)>0\) on \([0,1]\). By the classical method of elliptic regularization, the existence results are established.