含有对流项的奇异p-Laplacian问题的分歧 Alternative Title Bifurcation for singular p-Laplacian problems with convection term 张要星 Thesis Advisor 赵培浩 2010-05-25 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 分歧 对流项 p-Laplacian Abstract 本文研究如下p-Laplacian 方程 \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta_{p}u=g(u)+(p-1)\lambda^{\frac{1}{p-1}}|\nabla u|^p+\mu & \textrm{\text{在}$\Omega$\text{中}},\u>0 & \textrm{\text{在}$\Omega$\text{中}},\u(x)= 0 & \textrm{\text{在}$\partial\Omega$\text{上}} \end{array} % \right. \end{eqnarray*}% 的分歧问题.其中$\lambda,\mu \geq 0$,$\Omega$是$\textbf{R}^N$中的有界区域. 问题的奇异性是由非线性项g所引起的,其中g是一个严格递减函数,并且当$t\rightarrow 0$时,$g(t)\rightarrow\infty$.我们主要采用变量替换的方法以及p-Laplacian 的极大值原理和锥性质证明了:当 $\lambda(a+\mu)<\lambda_1$,上述问题有且只有一个正解;当 $\lambda(a+\mu)\geq\lambda_1$,上述问题没有正解.其中 $a=\lim_{t\rightarrow \infty}g(t)$,$\lambda_1$是$-\Delta_p$在$W_0^{1,p}(\Omega)$空间上的第一特征值. Other Abstract In this paper, we study the bifurcation problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p}u=g(u)+(p-1)\lambda^{\frac{1}{p-1}}|\nabla u|^p+\mu & \textrm{in $\Omega$},\\ u>0 & \textrm{in$\Omega$},\\ u(x)= 0 & \textrm{on$\partial\Omega$}, \end{array} % \right. \end{equation*}% where $\lambda, \mu \geq 0$, and $\Omega$ is bounded domain in $\textbf{R}^N$. The singular character of the problem is given by the nonlinearity g which is assumed to be strictly decreasing and unbounded around the origin. We use the skill to eliminate the conviction term and combine the maxmium principle and the Picone's identity for the p-Laplacian to prove that the above problem has a positive solution (which is unique) if $\lambda(a+\mu)<\lambda_1$； it has no positive solution if $\lambda(a+\mu)\geq\lambda_1$， where $a=\lim_{t\rightarrow \infty}g(t)$ and $\lambda_1$ is the first eigenvalue of $-\Delta_p$ in $W_0^{1,p}(\Omega)$. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225299 Collection 数学与统计学院 Recommended CitationGB/T 7714 张要星. 含有对流项的奇异p-Laplacian问题的分歧[D]. 兰州. 兰州大学,2010.
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