Asymptotic estimates of large solutions to the infinity Laplacian equations

The article is intend to study the asymptotic behavior of the large solutions near the boundary to the following infinity Laplace equation \(\triangle _{\infty }^{h} u =b(x)f(u), \ x\in \Omega ,\ u|_{\partial \Omega }=+\infty ,\) ${▵}_{\infty }^h{u=b(x)f(u),x\in \mathrm{\Omega },u|}_{\partial \mathrm{\Omega }}=+\infty ,$ where \( \triangle _{\infty }^{h}u:= |Du|^{h-3}\langle D^{2}uDu, Du \rangle \) ${▵}_{\infty }^{h}u:={|Du|}^{h-3}⟨D^{2}uDu,Du⟩$ for all \(h>1,\) $h>1,$ \(\Omega \) $\mathrm{\Omega }$ is a bounded domain with smooth boundary in \(\mathbb R^N\) ${\mathbb{R}}^N$ , \(b \in C(\bar{\Omega })\) $b\in C(\overline{\mathrm{\Omega }})$ which is positive in \(\Omega ,\) $\mathrm{\Omega },$ and \(f\in C^{1}(0,\infty )\) $f\in C^1(0,\infty )$ is positive increasing. The main feature of this paper is that the nonlinearity f is regularly varying at infinity with the critical index h.