两类分数阶非局部变分问题研究

In this project the variational methods and critical point theory are applied in the study of the existence and multiplicity of solutions for two classes of fractional non-local problems as well as the the analystic, geometric and topological properties of the solutions. First of all, we will study the existence and multiplicity of solutions for a class of fractional Shrödinger-Poisson system. And we will also explore the relationship between the solutions and potential functions or coefficients. Secondly, we will study a class of Kirchhoff type problems involving fractional Laplacian operator via the variational methods. We will focus on the case that the nonlinear term doesn’t satisfy the Ambrosetti-Rabinowitz condition. And we will also study some properties of the solutions.. The topics of this project are the new problems that we encountered in our recent scientific research. 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.

本项目将应用变分法和临界点理论研究两类分数阶非局部问题非平凡解的存在性和多重性以及解的拓扑、几何和分析性质。研究内容包括：(1)分数阶Shrödinger-Poisson系统，我们将利用非线性分析和变分法中的各种方法来研究分数阶Shrödinger-Poisson系统非平凡解的存在性和多重性并探索解与位势函数或系数之间的关系；(2) 带有分数阶Laplacian算子的Kirchhoff型问题，我们将重点研究非线性项不满足Ambrosetti-Rabinowitz条件时非平凡解的存在性和无穷多解的存在性，以及解的若干性质。. 本项目所选课题是我们在近年来研究中遇到的新问题，具有重要的理论意义和研究价值。我们期望通过本课题的研究，推进非线性分析和变分法理论与应用的发展。

项目组以分数阶Kirchhoff型方程和Shrödinger-Poisson系统为基本研究模型，综合应用变分法和非线性分析方法研究这两类方程在不同假设条件下非平凡解的存在性和多重性。基于分数阶椭圆问题与经典椭圆问题之间具有紧密的联系，我们研究了一些与两类分数阶非局部问题相关的其他方程或者方程组非平凡解的存在性和多重性问题，项目基本达到了预期目标，获得了一些成果，提升了项目组成员的科研水平，为后续进一步研究打下了基础。