分数阶非线性薛定谔方程解的存在性与集中性研究

In recent years, considerable attention has been given to Schrödinger equation driven by the fractional Laplacian. One of the reasons for this comes from fractional quantum mechanics in the study of particles on stochastic fields modeled by Lévy process. However, these equations cannot be treated well so far by use of the standard theory on the partial differential equations. This project is to investigate the existence, multiplicity, concentrations and multi-bump solutions by using variational and reduction methods. Our main contents includes: (1)We will research positive solutions of fractional nonlinear Schrödinger equations with potentials vanishing and unbounded at infinity, and subsequently establish some integral estimates on the solution, then derive the concentrations; (2) We will use Lyapunov-Schmidt reduction method to obtain the multi-bump solutions for fractional nonlinear Schrödinger equations, and imply the relation of interactions with bumps to the maximum or minimum point of potentials; (3) We will study the existence, concentrations of positive solution and the existence, multiplicity of sign-changing solution. It’s also our aim that to solve some important problems for fractional nonlinear Schrödinger equations, and develop some related theory, and provide some new ideas to find solutions for related fractional nonlinear elliptic equation.

分数阶薛定谔方程起源于量子力学中研究由Lévy过程所驱动的随机场中的粒子问题。目前，关于其解的相关研究已成为非线性分析领域的热点问题之一，但该类方程的偏微分方程理论尚不完善。本项目致力于采用变分理论和约化等方法研究分数阶非线性薛定谔方程, 给出解的存在性、多重性、集中性及多峰解，包括：（1）研究具有多个衰减或无界势的方程正解的存在性，建立解的积分估计，刻画解的集中性；（2）利用Lyapunov-Schmidt变分约化方法研究方程的多峰解，揭示峰与峰之间相互影响的多峰解的存在性与势函数极值点之间的关系；（3）研究临界情形下，方程正解的存在性、集中性及变号解的存在性和多重性。本项目旨在解决分数阶非线性薛定谔方程解的相关研究中的一些重要问题，发展和完善该领域的理论研究，为分数阶非线性椭圆方程中其它相关问题的研究提供新的思路。

分数阶偏微分方程不仅具有广泛的物理和应用背景,而且在数学上也具有重要的理论意义。本项目主要对几类分数阶椭圆型方程解的存在性、多重性及集中性进行了研究。具体来说：我们研究了势函数在几种不同条件下分数阶薛定谔方程解的存在性、多重性、集中性及衰减性；研究了具有Hardy位势的分数阶椭圆方程在临界情形下解的存在性和多重性；研究了一类具Hardy-Littlewood-Sobolev临界指数的分数阶椭圆系统非平凡解的存在性；研究了一类分数阶Choquard方程，证明了正解的存在性、对称性与不存在性。此外，还研究了一类Kirchhoff方程解的存在性及集中性。通过本项目的研究，可以帮助人们更好的理解应用学科中出现的分数阶偏微分方程。