Existence And Concentration Of Sign Changing Solutions For Fractional Schr(?)dinger Equation

The fractional order Schr(?)dinger equation has an important physical background and is a research focus in the field of nonlinear analysis in recent years.Different to the classical Laplacian operator,the fractional order Laplacian operator is nonlocal,and one cannot directly apply the classical elliptic partial differential equation theory to the fractional order Schr(?)dinger equation.In this thesis,we first introduce the physical background of the fractional order Schr(?)dinger equation and the research results closely related to this thesis.Secondly,we study a class of fractional order Schr(?)dinger equations with critical exponent.Under some assumptions on the weight functions,we obtain the existence of positive solution and sign changing solution by using the variational method.In order to overcome the difficulty of nonlocality of the fractional Laplacian operator,we use the harmonic extension method developed by L.caffarelli and L.Silvestre to transform fractional Schr(?)dinger equation into a local problem.Finally,we study the semiclassical problem of a class of fractional Schr(?)dinger equation,and prove that the sign changing solution exists and concentrates around the local minimum of the potential when the parameter tends to 0.