Boundary Behavior of Large Solutions to the Infinity Laplace Equations on the Half-Line

By adopting the method of upper and lower solutions, this article shows the blow-up rate of the unique nonnegative viscosity solution $l(t)$ of the boundary value problem
\begin{equation*} (u'(t))^{2}u''(t) =b(t)f(u(t)), \quad u(t)>0, \quad t>0, \qquad u(0)=\infty, \quad u(\infty)=0, \end{equation*}
where $b\in C^{1}(0,\infty)$, which is positive and nondecreasing on $(0,\infty)$ (and may vanish at zero).