We have $$L=r\sqrt{1+u_r^2}$$
Taking the derivatives gives,$$\frac{\partial{L}}{\partial{u}} = 0\\ \frac{\partial}{\partial{r}}\frac{\partial{L}}{\partial{u_r}} = \frac{\partial}{\partial{r}}(\frac{ru_r}{\sqrt{1+u_r^2}})$$
So the Euler_Lagrange differential equation becomes,
$$\frac{\partial{L}}{\partial{u}} - \frac{\partial}{\partial{r}}\frac{\partial{L}}{\partial{u_r}} = -\frac{\partial}{\partial{r}}(\frac{ru_r}{\sqrt{1+u_r^2}}) = 0\\\frac{ru_r}{\sqrt{1+u_r^2}} = a\\r^2u_r^2 = a^2(1+u_r^2)\\u_r^2 = \frac{a^2}{r^2-a^2}\\u(r) = a \int \frac{1}{\sqrt{r^2-a^2}} dr +b\\u(r) = a\cosh^{-1}(\frac{r}{a}) + b$$
