Study On The Related Problems Of Fractional Laplacians Equation

This paper is concerned with the related problems of fractional order laplacians op-erator,which come from practical problems of different fields,such as financial market is-sues,phase transformation,anomalous diffusion,semipermeable membrane,minimal surface problem.The solution to these problems requires the application of variational principle,the critical principle,fractional laplacians equation and so on.Therefore,the equation of frac-tional laplacians operator is of great importance to the study of modern natural phenomena.Fractional laplacians operator is a class of nonlocal elliptic operator,and widely applied in physical phenomena,abnormal physical phenomena and remote applications.This paper is to prove the existence of nontrivial solution and radial symmetric solution to a class of elliptic equations.The first chapter is an introduction,which briefly introduces the background and sig-nificance of the problem and the development process and the current situation of the re-search.At the end of this paper,we introduce the basic theorems,propositions and main work of this paper.In the second chapter,the existence of nontrivial solutioin to the problem is discussed.Where N?2,s ?(0,1),(-?)s is fractional laplacians operator,f:R~N×R?R is super-linear and has sub-critical growth.The conditions under which exist the nontrivial solution,the existence theorem and proof of nontrivial solution are also provided in this chapter.In the third chapter,the proof process of the existence theorem of the solution to radial symmetry under the subcritical condition is presented.The existence theorem of the solution to radial symmetry can be proved under the subcritical condition.Then,the critical point of the equation can be derived from the application of Nehari manifold and variational method in the space composed of radial symmetric functions.The critical point is the symmetric solution of the equation.