一类含测度的非强制拟线性椭圆方程解的存在性和不存在性

Existence and Non-Existence of Solutions to Some Degenerate Coercivity Quasilinear Elliptic Equations with Measure Data

中文关键词:  椭圆方程  非强制性  测度  存在性  不存在性

英文关键词:elliptic equation  degenerate coercivity  measures data  existence  non-existence

中文摘要:

      本文主要研究如下含非线性梯度项的非强制拟线性椭圆方程\begin{equation*}\left \{\begin{array}{rl}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.\end{equation*} 弱解的存在性和不存在性, 其中$\Omega\subseteq\mathbb{R}^N(N\geq3)$ 是有界光滑区域, $1

英文摘要:

      In this article, we study the existence and non-existence of weak solutions to the following quasilinear elliptic problem with principal part having degenerate coercivity and nonlinear term involving gradient, $$\left \{\begin{array}{ll}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.$$ where $\Omega\subseteq\mathbb{R}^N~(N\geq3)$ is a bounded smooth domain, $1