带梯度项的椭圆型方程边界爆破解的高阶估计

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

中文关键词:  二阶估计  三阶估计  半线性椭圆方程

英文关键词:second order estimates  third order estimates  semilinear elliptic equations

中文摘要:

      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.

英文摘要:

      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.