近年来，在物理、材料等学科研究中，导出了大量分数阶微分算子，特别是分数阶Laplacian算子引起了许多学者如Caffarelli院士等的极大兴趣并成为研究热点，但其非局部性也导致了许多本质的困难。2018年，张平文院士等提出了回火（tempered）分数阶Laplacian算子概念。在此基础上，本项目推广该算子到新的回火分数阶p-Laplacian算子，继而研究一类新的回火分数阶p-Laplacian算子的非线性问题，包括 a)运用现代分析技巧推导若干不等式，克服该算子奇异积分核的非线性、非局部特性，建立若干最值原理。b)克服该算子的退化性与奇异性，通过构造恰当函数，给出边界估计。最后，基于以上结果，利用直接移动平面法，研究一类回火分数阶p-Laplacian算子非线性问题在不同区域上正解的存在性、非存在性、单调性以及径向对称性。

Recently, in the research of physics, materials science and other disciplines, a large number of fractional differential operators have been derived, in which, fractional Laplacian operator has aroused the great interest of many scholars such as Academician Caffarelli and is becoming a hot research topic. Many essential difficulties are caused by the non-locality of fractional Laplacian operator. In 2018, Academician Pingwen Zhang et.al. proposed a new fractional tempered Laplacian operator and studied the numerical solution to a class of fractional tempered Laplacian problem. Based on above all, the project generalizes this operator to a new tempered fractional p-Laplacian operator, and then studies a series of nonlinear problem involving this new tempered fractional p-Laplacian operator..a) By using modern analytical techniques and various inequalities to overcome the nonlinearity and nonlocality of the singular integral kernel of the operator, we establish some maximum principles involving tempered fractional p-Laplacian operator..b) Due to the degeneracy (p>2) and singularity (p<2) of the operator, the narrow region principle applied in the direct moving plane method is no longer valid. In this project, by constructing appropriate functions, the boundary estimation of tempered fractional p-Laplacian operator is given to replace the narrow region principle. .At last, based on above all, the existence, nonexistence, radial symmetry and monotonicity of positive solutions to a series of nonlinear problems involving the tempered fractional p-Laplacian in different regions are studied by using the method of moving planes.