Existence Of Solutions Of A Class Of Fractional Laplacian Equations With Sublinearity

This paper deals with the positive global bounded solution of the semilinear frac-tional Laplacian equation,?-??^?/2u=??x,u?in Rⁿ.is the fractional order Laplace operator with 0<?<2.We use the global bounded solution U of the linear equation?-??^?/2U=??x?in Rⁿ,to construct the upper solution and lower solution of the original equation,which follows the existence of the solution.The maximum principles in the whole space and the method of upper and lower solution are the main tools in our research,especially in the study of the uniqueness of the solution and conditions of the existence of entire large solutions for equation?-??^?/2u=-??x?f?u?in Rⁿ.This paper is divided into four chapters:In the first chapter,some backgrounds and introductions of the problem are presented;Chapter 2 contains several definitions of the fractional Laplace operator which are proved equivalent as follows,and the pre-cise definitions of the weak solution with respect to our Laplacian equations.We also present the main methods that we use in the next chapter;Chapter 3 covers our main results in this study.First,we focus on the existence of the global bounded solution of the linear equation,which is proved by the method of upper and lower method as follows.Secondly,we give the proof of our theorem and generalize our existing con-clusions.Chapter 4 consists of the application of the uniqueness of the solution of?-??^?/2u=-??x?f?u?in Rⁿand the study of existence of the entire large solution.