Higher Order Estimates for Boundary Blow-Up Solutions of Elliptic Equations with Gradient Term

DOI：10.3770/j.issn:2095-2651.2021.02.005

 作者 单位 张亚杰 宁波大学数学与统计学院, 浙江 宁波 315000 马飞遥 宁波大学数学与统计学院, 浙江 宁波 315000 沃维丰 宁波大学数学与统计学院, 浙江 宁波 315000

本文系统地研究了有界光滑域中带有梯度项的半线性椭圆型方程的边界爆破解的高阶渐近性质. 推导了方程的二阶和三阶边界行为. 结果表明,边界平均曲率及其梯度在解的高阶渐近展开中的作用.

In this paper, the higher order asymptotic behaviors of boundary blow-up solutions to the equation $\Delta\,u={u}^{p}\pm |\nabla u|^{q}$ in bounded smooth domain $\Omega \subset {R}^{N}$ are systematically investigated for $p$ and $q$. The second and third order boundary behaviours of the equation are derived. The results show the role of the mean curvature of the boundary $\partial \Omega$ and its gradient in the high order asymptotic expansions of the solutions.