临界点理论是讨论微分方程解的存在性的重要工具，但目前在分数阶拉普拉斯方程周期解的存在性方面还未见有结果。主要原因是由于分数阶拉普拉斯算子的奇异性和非局部性, 使得它与经典的拉普拉斯算子有着很大的不同。这就为利用临界点理论研究分数阶拉普拉斯方程周期解增添了困难。本项目将运用临界点理论研究分数阶微分方程周期解的存在性和多解性，在非线性项满足次线性,超线性,渐近线性等情形下讨论解的存性、多解性以及正解的存在性。这些研究将进一步改进和完善分数阶微分方程的基本理论，并且为某些实际问题的解决提供新的思路、方法和重要的理论依据。

Critical point theory is an important tool to discuss the existence of solutions for differential equation, but there is few result on the existence of periodic solutions for fractional Laplace equation, because the singular and nonlocality of the fractional differential operator make many difficulties for the study of fractional differential equations. This subject will study the existence and multiplicity of solutions and the existence of positive solutions when the nonlinearity is linearity, superlinearity and sublinearity. These study will improve the basic theory of fractional differentional equation and provides new methods for solving some practical problems.