分数阶偏微分方程已出现在物理、力学和生物材料等应用学科，应用研究促进了相应的数学理论发展。目前为止，低阶分数阶方程已经有丰富的研究结果，而高阶分数阶方程则因高阶算子定义结构的复杂性而结果甚少。本项目致力于研究高阶分数阶拉普拉斯方程解的存在性、不存在性与对称性等，具体研究内容包括：发展高阶分数阶拉普拉斯算子的极值原理，创建直接针对高阶分数阶拉普拉斯方程的移动平面法；降低算子的阶数，在不同的空间中寻找算子的复合，探求复合形式的Green函数及其性质，证明高阶调和函数的Liouville定理，然后借助需要研究的微分方程与积分方程的等价性结果，建立高阶分数阶拉普拉斯方程的积分形式的移动平面法；应用所建立的两种移动平面法，研究几类高阶分数阶拉普拉斯方程解的存在性等，如研究分数阶Lane-Emden猜想问题。本项目所建立的研究方法将有效推进高阶非局部伪微分算子理论的研究进展。

Fractional partial differential equations have appeared in many applications, such as physics, mechanics, biological materials, etc. The application research promotes the development of corresponding mathematics theory. Until now , there are a series of fruitful results about lower order fractional equation, while higher order fractional equation was much less studied for the complicated structure of definition. The project is devoted to study the symmetry, existence and nonexistence of the solutions for equations with higher order fractional Laplacian. The specific ideas are that: develop the Maximum Principles for higher order fractional Laplacian, and thus establish a direct method of moving planes for higher order fractional Laplacian; Using the composition of the fractional Laplacian, construct Green’s functions for the composition and study its properties. Prove the Liouville theorem for higher order harmonic function and obtain the equivalence of differential equation and integral equation. Hence, form the method of moving planes in integral forms for higher order fractional Laplacian operator. Based on the two methods of moving planes, the project will continue to consider the properties of solutions for equations with higher order fractional Laplacian, such as the fractional Lane-Emden Conjecture. The systematical methods which will be introduced in this project will stimulate the development of the higher order nonlocal pseudo-differential operator area.