分数阶Schrödinger-Poisson系统的变分方法研究

Nowadays, variational method research of elliptic equations (systems) involving fractional Laplace operator is one of the hot topic internationally. Becase the fractional Laplace operators appear in the mathematical model of variuous physical phenomenon, such as, anomalous diffusion, water wave, molecular dynamics, quantum mechanics and so on. On the other hand, Since the fractional Laplace operator is of nonlocal operators, some classical methods and techniques in the study of elliptic equations (systems) may not be applied. Hence, it is theoretically important in the study of elliptic equations (systems) involving the fractional Laplace operator. In this project, we shall study the existence and multiplicity of solutions for fractional Schrödinger-Poisson system by using the variational methods. We will study this system from the following three aspects:.(1) The existence of ground state solutions will be studied by constrained minimization arguments and pertubation variational methods..(2) The existence of infinitely many solutions will be studied by fountain theroem and invariant sets of descending flow..(3) The existence of infinitely many nonradial positive solutions and the existence and multiplicity of bump solutions will be studied by variational reduction methods.

目前，带分数阶Laplace算子的椭圆型方程（组）的变分方法研究是国际上的热点问题之一。由于各种各样的物理现象的数学模型需要使用分数阶Laplace算子，比如，反常扩散、水波、分子动力学、量子力学等。另一方面，由于分数阶Laplace算子是非局部算子，致使传统的研究带分数阶Laplace的椭圆型方程（组）的一些方法和技巧可能不适用。因此，研究带分数阶Laplace算子的椭圆型方程（组）具有重要的理论意义。本项目拟采用变分方法研究分数阶Schrödinger-Poisson系统解的存在性和多重性。本项目将从下面三个方面进行研究：.（1）采用约束极小化方法和扰动变分方法研究其基态解的存在性；.（2）采用喷泉定理和下降流不变集方法研究其无穷多解的存在性；.（3）采用变分约化方法研究其无穷多非径向正解和峰值解的存在性和多重性。

本项目主要研究了下述问题：1)研究了次临界、带扰动项的临界分数阶Schrödinger-Poisson系统基态解的存在性；2) 研究了带竞争位势的分数阶Schrödinger-Poisson系统基态解的存在性和集中性；3) 研究了带竞争位势、消失位势和渐近周期位势的分数阶Schrödinger方程基态解、正解的存在性；4) 研究了带p-Laplacian,(p,q)-Laplacian等拟线性椭圆型方程解的存在性、多重性和集中性；5）研究了临界或超临界的Kirchhoff型方程基态解、正解的存在性。在本项目的执行期间，共发表论文17篇，还完成5篇已投稿的论文。