Boundary behavior of large solutions to elliptic equations with nonlinear gradient terms

Abstract

In this paper, we mainly study the asymptotic behavior of solutions to the following problems \({\triangle u \pm a(x)| \nabla u|^{q} = b(x)f(u), x \in \Omega, \ u|_{\partial \Omega} = + \infty}\), where Ω is a bounded domain with a smooth boundary in \({\mathbb{R}^{N} (N \geq 2)}\), q > 0, \({a \in C^{\alpha}(\bar{\Omega})}\) is positive in Ω, and \({b \in C^{\alpha}(\bar{\Omega})}\) is nonnegative in Ω and may be vanishing on the boundary. We assume that f is Γ-varying at ∞, whose variation at ∞ is not regular. Our analysis is based on the sub-supersolution method and Karamata regular variation theory.