分数阶椭圆方程与积分方程的若干问题

In this proposal, we will investigate the existence and properties of solutions for fractional elliptic equations(systems) and that of integral equations(systems). Firstly, we are concerned with the existence and the qualitative properties of the solutions of fractional elliptic equations(systems), including the existence of radial solution, non-radial solution, asymptotic behavior of solution for fractional elliptic equations with Henon weights; the existence and multiplicity of solutions for critical and double critical problems with Hardy weights, partial Hardy weights when the singularity point on the boundary of domain. The effect of geometry and topology of domain to the existence and the number of solution. Secondly, we will show the existence and the qualitative properties of the solutions of integral equations(systems) associated with fractional elliptic operators. We establish some weighted Hardy-Littlewood-Sobolev inequalities associated with several potential type operators and discuss the symmetry, momotonicity and Holder continuous by using the moving planes methods and regularity lifting theorems.

本项目研究分数阶椭圆方程（组）与积分方程（组）解的存在性和解的性质。首先讨论不同形式的分数阶椭圆方程（组）解、多解的存在性以及解的其它性质，包括分数阶Henon型方程（组）径向及非径向正解的存在性以及解的渐进行为；奇点在边界的带Hardy权、部分Hardy权的临界、双临界问题解及多解的存在性；区域的拓扑和几何结构对解的影响。其次，考虑与分数阶椭圆算子相关的积分方程（组）解的存在性以及解的其它性质，利用打靶的方法研究解的存在性；先建立与位势算子相关的HLS不等式，再利用积分方程的移动平面法和正则性提升法得到积分方程（组）解的对称性、单调性、Hölder连续性。

本项目首先研究了一类极小化问题，证明了此极小化问题可达，还得到了达到函数的衰减性。讨论分数阶Henon型方程（组）解的存在性、解的渐进行为。其次证明了与一类高阶Schrodinger算子相关的Riesz位势算子在Lp空间上的有界性，做为应用得到了与Riesz位势核的相关的积分方程解的Lp估计。此外还研究了由Riesz位势算子生成的交换子在Lp空间、BMO空间上的有界性，同时还得到了该位势算子在Hardy空间上的有界性。最后，得到了一类多线性平方函数和Lipschitz函数生成的交换子在乘积Lebesgue空间上的有界性，同时证明了乘积Lebesgue空间到Lipschitz空间，乘积Lebesgue空间到BMO空间、乘积Lebesgue空间到Triebel-Lizorkin空间上的映照。我们还建立了一些次线性积分算子和加权Lipschitz函数、加权BMO函数生成的交换子在加权Morrey空间上有界的判别准则。