几类分数阶非线性椭圆方程解的存在性与集中性

发布时间：2018-05-04 08:29

本文选题：分数阶拉普拉斯算子 + 临界指数　；参考：《南京师范大学》2015年博士论文

【摘要】：分数阶拉普拉斯问题可以用来描述物理学、生物学、化学、金融经济、概率等领域中的许多重要现象.特别地,在概率的观点下,分数阶拉普拉斯算子被视为稳态Levy扩散过程的无穷小生成元.因此,分数阶拉普拉斯微分方程解的相关问题研究目前已成为非线性分析领域的热门研究方向之一在本论文中,我们利用非线性分析中的临界点理论和变分约化等方法研究了两类具有临界指数的分数阶椭圆方程解的存在性、多重性及分数阶非线性Schrodinger方程解的存在性和集中性,获得了一系列新的结果.具体包含以下四章内容：在第一章中,我们利用Nehari流形方法和Ljusternik-Schnirelmann筹数理论研究了一类具有临界指数的分数阶非线性Schrodinger方程.证明了方程在两种不同情形下具有基态解和catΛδ(Λ)个非平凡解.在第二章中,我们利用变分扰动方法研究了分数阶非线性Schrodinger方程解的存在性及集中性.设合理的假设下,证明了所得解集中在函数г(x)的临界点.我们所得结果推广了文献[36]和[44]的结果.在第三章中,我们研究了一类分数阶非线性椭圆方程的多峰解,其中Q(x)为正的连续有界函数.利用 Lyapunov-Schmidt变分约化方法得到,对任意的正整数七,方程具有一个七-峰的正解,且其集中在Q的严格局部极小点处.我们把文献[65]的结果推广到了分数阶情形.最后,我们利用调和扩展技术和临界点理论,研究了一类具有临界指数的非齐次分数阶Laplacian司题,证明了此类问题至少具有两个正解.同时,在一类线性正型区域上,我们获得了一个正解的不存在性结果.此结论推广了文献[85]中的不存在性结果.
[Abstract]:Fractional Laplace problem can be used to describe many important phenomena in physics, biology, chemistry, financial economy, probability and so on. In particular, in the view of probability, fractional Laplace operator is regarded as infinitesimal generator of steady-state Levy diffusion process. Therefore, the research on the solutions of fractional Laplacian differential equations has become one of the hot research directions in the field of nonlinear analysis. By using the critical point theory and variational reduction in nonlinear analysis, we study the existence, multiplicity, existence and centrality of solutions for two classes of fractional elliptic equations with critical exponents. A series of new results were obtained. In the first chapter, we study a class of fractional nonlinear Schrodinger equations with critical exponents by using Nehari manifold method and Ljusternik-Schnirelmann number theory. It is proved that the equation has a ground state solution and a nontrivial solution of cat A 未 (A) in two different cases. In chapter 2, we study the existence and centrality of solutions of fractional nonlinear Schrodinger equations by using variational perturbation method. Under reasonable assumptions, it is proved that the solution is concentrated at the critical point of the function. Our results extend the results of references [36] and [44]. In chapter 3, we study the multimodal solutions of a class of fractional nonlinear elliptic equations, where QX) is a positive continuous bounded function. By using the Lyapunov-Schmidt variational reduction method, it is obtained that for any positive integer 7, the equation has a positive solution of a seven-peak, and it is concentrated at the strictly local minimum of Q. We extend the results of [65] to the fractional order case. Finally, by using harmonic expansion technique and critical point theory, we study a class of nonhomogeneous fractional Laplacian problems with critical exponents, and prove that these problems have at least two positive solutions. At the same time, we obtain a nonexistence result of positive solutions in a class of linear positive domains. This conclusion generalizes the nonexistent results in [85].
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175.25

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