Infinitely Many Solutions For Nonlinear Schr???dinger Equations And Systems In R^N

In this thesis,by using the methods of perturbation,truncation and invariant sets of descending flow,we obtain the existence of infinitely many solutions of a modified nonlinear Schr???dinger equation,and get the existence of infinitely many sign-changing solutions of nonlinear Schr???dinger systems.This thesis is divided into three chapters,the main contents are as follows.In Chapter 1,we give the research problem and its backgrounds,and give our main results.In Chapter 2,we consider the following modified nonlinear Schr???dinger equation-?u + u-u?u²= a?x?|u|^q-2u,x ? R^N,where N ? 3,4 < q < 2 · 2*=?4N?/?N-2?,and the potential function a?x?is positive,bounded and verifies suitable decay assumptions.The equation only has a variational structure formally,there is no suitable space in which the functional enjoys both smoothness and compactness properties.By adding a 4-Laplacian operator and using the methods of perturbation and truncation,we obtain the existence of infinitely many solutions for this problem.In Chapter 3,we consider the following nonlinear Schr???dinger systems where ?₁,?₂are positive constants,and ?_ij,i,j = 1,2 satisfies suitable assumptions.We consider two cases that the matrix(?_ij?x?),i,j = 1,2 is positive definite and is not positive definite.By using the methods of perturbation and invariant sets of descending flow,we obtain the existence result of infinitely many sign-changing solutions.