一类加权非线性椭圆型方程大解的渐近行为

摘要/Abstract

摘要：

该文研究加权非线性椭圆型问题 $ \Delta u(x)= a(x)f(u(x))+ b(x)|\nabla u(x)|^q$, $x\in \Omega$, $u|_{\partial\Omega}=+\infty $, 解的整体和边界渐近行为. 其中, $\Omega$ 是 $\mathbb R^n$ ($n\geq2$) 中的有界光滑区域, $q\in (0, 2]$, $f(s)=s^p$ ($p>0$), 或者 $f(s)=e^s$, $a, b\in C^\alpha(\Omega)$ 在 $\Omega$ 上是正的, 但允许在边界 $\partial \Omega$ 退化到 $0$ 或具有适当的奇性. 在 $a$ 和 $b$ 满足适当的条件下, 给出了解的完整分类.

关键词: 半线性椭圆型方程, 非线性梯度项, 权函数, 大解, 渐近行为

Abstract:

The paper is mainly concerned with global and boundary asymptotic behavior of classical large solutions to semilinear elliptic equation $\Delta u(x)= a(x)f(u)+ b(x)|\nabla u|^q$, $x\in \Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb R^n$ with $n\geq 2$, $q\in (0, 2]$, $f(s)=s^p$ with $p>0$, or $f(s)=\exp s$, $a, b\in C^\alpha(\Omega)$ which are positive and coupling in $\Omega$, but may vanish or blow up on the boundary properly. A complete classification of solutions is given under the appropriate conditions on $a$ and $b$.

Key words: semilinear elliptic equation, nonlinear gradient terms, weights, large solutions, asymptotic behavior

中图分类号:

O172.2

张嘉洋. 一类加权非线性椭圆型方程大解的渐近行为[J]. 数学物理学报, 2026, 46(5): 1825-1836.

Jiayang Zhang. Asymptotic Behavior of Large Solutions for a Class of Weighted Elliptic Equations[J]. Acta mathematica scientia,Series A, 2026, 46(5): 1825-1836.

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