| 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 Title | Multiple positive solutions for a class of quasilinear elliptic equations with nonlinear Neumann boundary condition
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| 代丽芳 |
| Thesis Advisor | 赵培浩
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| 2009-06-03
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
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| Keyword | p-Laplacian 方程
正解
(PS)条件
变分法
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| 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 Abstract | In 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
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/224482
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| Collection | 数学与统计学院
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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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