兰州大学机构库 >数学与统计学院
含有对流项的奇异p-Laplacian问题的分歧
Alternative TitleBifurcation 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 AbstractIn 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
Identifierhttps://ir.lzu.edu.cn/handle/262010/225299
Collection数学与统计学院
Recommended Citation
GB/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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