【作者】 赵辉；

【导师】 付永强；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2006， 硕士

【摘要】 本文的主要研究内容是在空间L^{p（ x）}和W^{k, p（x）}的基本理论体系的基础上,研究p （x）-Laplacian问题多重解的存在性。随着弹性力学的发展,对非标准增长条件p （x）-Laplacian问题的研究是近年来发展起来的一个新的研究课题。p （x）-Laplacian方程来源于许多物理背景,例如,非Newton流体问题（Newton流体问题对应于p =2）,非线性弹力问题等。因此对这类问题的研究具有广泛的理论与实际意义。对p （x）?Laplacian问题的研究,有很多不同的方法。近期,临界点理论似乎成为解决偏微分方程问题的一个非常有用的工具。利用这个工具可以成功地解决不少微分方程解的存在性问题,尤其是具有非标准增长条件的Laplacian问题。本文借助广义Lebesuge空间L^p（x）和广义Sobolev空间W^k,p（x）的基本理论,尤其是嵌入定理,研究了p （x）-Laplacian问题的多重解的存在性,这里Ω（?）R^N为具有光滑边界（?）Ω的有界区域, p （x）在Ω上Lipschitz连续并满足2 < p^-≤p（ x）≤p⁺<N, g是连续的。本文使用的方法是变分法且运用了临界点理论和Brezis-Nirenber定理。在研究过程中巧妙地使用了空间分割的技巧。得出如下结果:当g满足一定条件时, p （x）-Laplacian问题至少有两个非平凡弱解。更多还原

【Abstract】 The main content of this paper is to study the existence of multiple solutions of a p （x）-Laplacian problem, our study is on the base of the basic theory of the spaces L^p（x）and W^k,p（x）.With the development of elastic mechanics, the study of p （x）-Laplacian problem with nonstandard growth conditions is a new topic developed in recent years. The p （x）-Laplacian equation arises naturally in various contexts of physics, for instance, in the study of non-Newtonian fluids （the case of Newtonian fluid corresponding to p =2）, and in the study of nonlinear elasticity. So studying such problems has wide meaning of theory and practice. To study p （x）-Laplacian problem, we have various means. Recently, Critical Point Theory seems to be a very useful tool to solve partial differential equation. We can prove the existence of solutions of much partial differential equations successfully with this tool, especially for the Laplacian problem with non-standard growth conditions.In this paper, by means of the basic theory of generalized Lebesuge space L^p（x）and generalized Sobolev space W^k,p（x）, especially the Embedding Theorems, we study the existence of multiple solutions of the p （x）-Laplacian problemWhereΩ（?）RN is a bounded domain with smooth boundary （?）Ω. p （x）is Lip-schitz continuous onΩand satisfies 2 < p ^-≤p（ x）≤p⁺<N, g is continuous. Our approach is variational and uses Critical Point Theory and Brezis-Nirenberg Theorem. In the process of studying, we use the trick of cutting space skillfully.We get the following result: when g satisfies certain conditions, the p （x）-Laplacian problem has at lest two nontrivial weak solutions.更多还原