几类分数阶非局部椭圆型方程的变分问题研究

In this project, we shall investigate several fractional important Schrodinger-Poisson systems，Kirchhoff equations and Choquard equations, possessing very important application background. Under variant local conditions imposed on the potential V, we study the existence, multiplicity, concentration and decay properties of positive solutions to these equations, by means of variational and topological methods. We study the existence and asymptotical behavior of ground state solutions to these equations with potential wells. The existence of sign-changing solutions for these equations with zero potentials, are also considered. We study the existence and multiplicity of solutions for singular Kirchhoff equations with negative exponent and the bifurcation phenomena of solutions. Using concentration-compactness principle and Brezis-Nirenberg technique, we study the existence of solutions for Choquard equations with Hardy-Littlewood-Sobolev exponent.

在本项目中，我们将对几类具有重要应用背景的分数阶Schrodinger-Poisson系统，Kirchhoff 方程和Choquard方程开展研究。在位势V满足不同的局部条件下，运用变分与拓扑方法研究正解的存在性、多解性、集中性及衰减性等问题。研究具有位势井情形下上述方程基态解的存在性和渐近行为。在具有零质量位势V情形下，研究变号解的存在性。对于具有负指数的奇性Kirchhoff方程，研究解的存在性和多重性，解的分歧现象。用集中紧性准则和Brezis-Nirenberg技术研究带有临界Hardy-Littlewood-Sobolev指数型增长的Choquard方程解的存在性。

在本项目中，我们对几类具有重要应用背景的分数阶Schrodinger-Poisson系统，Kirch.hoff 方程和Choquard方程开展了研究。在位势满足不同的全局及局部条件下，运用变分与拓扑方法研.究正解的存在性、多解性、集中性及衰减性等问题。在具有衰减位势情形下，研究变号解的存在性。对于具有负指数的奇.性Kirchhoff方程，研究解的存在性和多重性，解的分歧现象。利用集中紧性分析和能量估计研究带有临界指数增长的分数阶Choquard方程解的存在性；.研究双临界指数增长Schrodinger-Poisson系统基态解的存在性。应用变分与几何分析方法对几类orlicz闵可夫斯基问题解的存在性与唯一性开展了研究。