Existence And Symmetry Of Solutions For Fractional Schr(?)dinger Equations Involving The Bessel Operator

The fractional operator of the fractional nonlinear Schr(?)dinger equations is nonlocal pseudo partial operator,whose nature and characteristics are different from the integral operator,which makes the fractional equation have a wider application in crystal dislocation,land circulation,optimization of physics and finance.By using variational method and moving planes method,this paper makes a study of existence and symmetry for fractional Schr(?)dinger equations associated with Bessel operator.Chapter 1 is the introduction,which mainly expounds the research background of fractional Schr(?)dinger equations,the research status on fractional Schr(?)dinger equation related to Bessel operator,as well as the important preparatory knowledge associated with this paper.Chapter 2 studies the fractional Schr(?)dinger equations which involve the Bessel operator as followThe potential functionV(x)is periodic,and the lowest bound is negative,which makes the energy functional be strongly indefinite.The existence of the ground state solution of the equation(1)is obtained by using the variational method and the concentration compactness principle.In Chapter 3,we study the fractional nonlinear equation related to Bessel operator in bounded domainIn the first part,we use the variational method to prove that(2)has nontrivial solutions in the nonlinear term satisfies(AR)condition of bounded area.By using moving planes method,the second part is proved the positive solutions of the equation are symmetric and monotonously decreasing when ? is symmetrical convex bounded area with respect to the some component.Especially,the positive solution of the equations sre radially symmetric and monotonously decreasing.