Let \(D \subset {\mathbb {R}}^{N}\) , ( \(N\ge 3\) ), be an exterior domain with compact \(C^{2}\) boundary and \(\delta (x)=dist(x,\partial D)\) . Our purpose in this work is to provide the exact boundary behavior of positive solutions to the following nonlinear value problem \(\begin{aligned} \left\{ \begin{array}{l} \Delta u=a(x)g(u), x\in D, u>0\text { in }D ,\\ \lim _{\delta (x) \longrightarrow 0} u=+\infty \text { and } \lim _{|x|\rightarrow +\infty } u=+\infty , \end{array} \right. \end{aligned}\) where \(g\in {\mathcal {C}}^{1}((0,\infty ),(0,\infty ))\) is nondecreasing and the function \(a\in {\mathcal {C}}_{loc}^{\gamma }(D)\) , \((0<\gamma <1)\) , satisfying for each \(x\in D\) , \(\begin{aligned} 0\!<\!b_{1} = \underset{\delta (x) \longrightarrow 0}{\lim }\inf \frac{a(x)}{(\delta (x))^{-\lambda }L_{1}(\delta (x))}\!\le \! \underset{\delta (x) \!\longrightarrow 0}{\lim }\sup \frac{a(x)}{(\delta (x))^{-\lambda }L_{1}(\delta (x))}\!=\!b_{2}\!<\!\infty \end{aligned}\) and \(\begin{aligned} 0<c_{1}= \underset{|x| \longrightarrow +\infty }{\lim }\inf \frac{a(x)}{|x|^{-\mu }L_{2}(|x|)}\le \underset{|x| \longrightarrow +\infty }{\lim }\sup \frac{a(x)}{|x|^{-\mu }L_{2}(|x|)}=c_{2}<\infty , \end{aligned}\) where \(\lambda \le 2\) , \(\mu \ge 2\) and \(L_{1},L_{2}\) are in class of Karamata functions.
