几类分数阶椭圆型问题的变分法研究

In this project we will study the existence and multiplicity of solutions for several fractional ellptic problems where the fractional p-Laplacian equation is the main prototype by the variational methods and critical point theory. As a classical topic in functional analysis and harmonic analysis, fractional function spaces and the corresponding nonlocal equations can be applied in different subjects, such as, the thin obstacle problem, optimization, finance, phase transitions, anomalous diffusion, materials science. By the variational methods in this project, first of all, we will consider the solvability and multiplicity for some quasilinear fractional elliptic problems with local growth conditions and study the properties of them; secondly, some weighted quasilinear fractional elliptic problems on the whole space will be considered, we will study the existence and multiplicity of solutions for them and some properties of the solutions. The topics of this project are included in the international frontiers of the scientific research of variational theory, and these problems have important theoretic meanings and research values. We hope to make some contributions to the development of nonlinear analysis via the study of this project.

本项目将主要以分数阶p-Laplacian方程为原型应用变分法和临界点理论来研究几类分数阶椭圆型问题的可解性和多解性。作为泛函分析和调和分析的经典议题之一，分数阶函数空间以及与其相关的非局部方程在很多领域内都有着很重要的应用，例如薄障碍问题、最优化问题、经济学、相变问题、反常扩散问题、材料科学等。在本项目中利用变分法，首先我们将考虑几类带有局部增长性条件的拟线性分数阶椭圆型问题的可解性和多解性以及解的若干性质；其次我们将研究几类全空间上加权拟线性分数阶椭圆型问题的可解性和多解性，并考虑解的各种性质。本项目所选课题是我们在近年来研究中遇到的新问题，具有重要理论意义和研究价值。我们期望通过本课题的研究，推进非线性分析理论与应用的发展。

在该项目执行期内，项目组以分数阶椭圆方程为主要研究对象，综合应用变分方法和非线性分析方法研究该类方程非平凡解的存在性、多解性。借助于分数阶椭圆方程解的某种先验估计，在非线性项只在零点附近加条件的情形下得到了有界区域上半线性、拟线性分数阶椭圆方程非平凡解的存在性和多解性。在位势函数满足某些可积性条件的前提下，通过构造适当的加权型Sobolev函数空间、建立Sobolev型嵌入定理，项目组研究了一类全空间上拟线性分数阶椭圆问题的多解性。通过计算临界群，利用Morse理论，项目组还研究了一类拟线性椭圆方程非平凡解的存在性和多解性问题。. 这些研究结果完成了项目申请中所涉及的主体内容，丰富了椭圆方程的理论。在项目执行期间，项目组已在国际SCI收录杂志发表学术论文3篇。