Existence And Concentration Of Ground State Solutions For A Class Of Fractional Schr?dinger Equations

In this thesis,by using the variational method,we prove the existence and concentration of ground state solutions for the following indefinite strongly fractional Schrodinger equations#12 where α ∈(0,1),ε is a positive parameter,N>2α,(-△)α stands for the fractional Laplacian.f:R→R is a continuous function with subcritical growth.The function A:RN→R is a positive continuous function and satisfies some appropriate assumptions.The potential function V:RN→R is a ZN-periodic continuous function.To begin with,we use a new minimax character that was first found by Szulkin and Weth,which combine the generalized Nehari manifold method to prove the existence of the ground state solution of the limit equation and the monotonicity of the ground state solutions with re-spect to the parameter.Then,by comparing the ground state energy between the limit equation and the original equation,we obtain the existence of the ground state solution of the original equation.Finally,we consider the concentration phenomenon of the ground state solutions when the parameters approach to zero under the existence of the ground state solutions.In this problem,we first obtain the strong convergence of the ground state solutions with respect to the parame-ters in the fractional Sobolev space Hα(RN).Then,by using Bessel kernel convolution with the nonlinear terms,we get the integral expression of the ground state solutions.By using the prop-erties of the Bessel kernel and the mathematical analysis,we estimate the decay of the ground state solution at infinity.Finally,we get the result that the maximum points of the ground state solutions is concentrated on the maximum points of the function A(x).