兰州大学机构库 >数学与统计学院
一类具非线性 Neumann 边值条件的拟线性椭圆方程的多重正解
Alternative TitleMultiple positive solutions for a class of quasilinear elliptic equations with nonlinear Neumann boundary condition
代丽芳
Thesis Advisor赵培浩
2009-06-03
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keywordp-Laplacian 方程 正解 (PS)条件 变分法
Abstract本文研究如下\ p-Laplacian 问题 - \triangle_p u + u^{p-1}=f(u)e^{u^\alpha} u>0 x\in \ \Omega, |\nabla u|^{p-2} \dfrac {\partial u}{\partial v}= \lambda \psi u^q x \in \partial\Omega, (1.1.1) 弱解的存在性, 其中\ $\Omega$ 是\ $\mathbb{R}^{2}$ 中具\ $C^2$ 边界的有界区域,$p>2,\,\alpha \in (0,2],\, \lambda >0,q \in [0,p-1)$, 且\ $\psi\geq 0$ 是\ $\bar{\Omega}$ 上的非负$H\ddot{o}lder$连续函数. $f(u)$ 是当\ $u\rightarrow \infty $ 时的一个多项式扰动.在对\ $f(u)$ 的适当假设条件下,运用变分原理我们得到: 存在常数$\Lambda \in (0, \infty)$ 使得当$\lambda \in (0,\Lambda)$时,问题 (1.1.1) 至少有两个解,$\lambda =\Lambda$时至少有一个解,而当$\lambda>\Lambda $ 时无解.
Other AbstractIn this paper, we consider the existence of weak solutions of the following p-Laplacian problem - \triangle_p u + u^{p-1}=f(u)e^{u^\alpha} u>0 x\in \ \Omega, |\nabla u|^{p-2} \dfrac {\partial u}{\partial v}= \lambda \psi u^q x \in \partial\Omega, (1.1.1) where $\Omega \subset \mathbb{R}^{2}$ is a bounded domain with $C^2$ boundary,$p>2,\,\alpha \in (0,2],\, \lambda>0, q \in [0,p-1)$, and $\psi\geq 0$ is an nonnegative $H\ddot Ho}lder$ continuous function on $\bar{\Omega}$. Moreover, $f(u)$ is a polynomial perturbation of $u$, when $u\rightarrow \infty $. Resorting to the variational method and some different hypothesis of $f(u)$, we show that there exists a constant\ $\Lambda \in (0, \infty)$ such that problem (1.1.1) has at least two solutions if $\lambda \in (0,\Lambda)$, at least one solution if $\lambda =\Lambda$ and no solution when $\lambda >\Lambda$.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224482
Collection数学与统计学院
Recommended Citation
GB/T 7714
代丽芳. 一类具非线性 Neumann 边值条件的拟线性椭圆方程的多重正解[D]. 兰州. 兰州大学,2009.
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