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 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 Keyword p-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 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 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224482 Collection 数学与统计学院 Recommended CitationGB/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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