Multiple Solutions For Several Classes Of Fractional Schr(?)dinger Equations

Along with science's and technology's development, various nonlinear problem has aroused people's widespread interest day by day, and so the nonlinear anal-ysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear analysis, it takes the nonlinear problems appearing in mathematics and physics as background to establish some general theories and methods to handle nonlinear problem. Because it can well explain various natural phenomenon, so, the mathematical world and the natural science world pay more attention to the nonlinear functional analy-sis. The nonlinear Schr(?)dinger equation stems from the applied mathematics, the physics and each kind of application discipline. It is one of the most active domains of the integral differential equation. They have obtained some new result for the Schr(?)dinger equations. The existence of solutions as well as multiplicity for a class of fractional Schr(?)dinger is also the hot spot which has been discussed in resent years.In this paper, we use symmetric mountain pass theorem, Nehari manifold and other critical point theory, to discuss the existence of multiple solutions for some special kinds of fractional Schr(?)dinger equation, and prove the existence of multiple solutions. The thesis is divided into three sections according to contents.Chapter 1 We introduce the background, fundamental knowledge such as s-pace and norm about fractional Schr(?)dinger equations.Chapter 2 We consider the fractional Schr(?)dinger equation without (A-R) condition: R is a continuous function. Here (-?)s is so-called fractional Laplacian operator of order ??(0,1). Potential V(x) is a continuous function in RN. Under appropriate conditions, we use symmetric mountain pass theorem to prove the existence of multiple solutions for this equation.Chapter 3 We consider the fractional Schr(?)dinger equation with unbounded potential function: changes sign in RN. The unbounded potential V(x) is a continuous function in RN. Under appropriate conditions, we use Nehari manifold to prove the existence of multiple solutions for this equation.