Studies On The Existence Of Solutions For Two Kinds Of Fractional Schr(?)dinger Equations By Variational Methods

With an important physical background,the fractional Schr(?)dinger equation has attracted great attention from the field of nonlinear analysis in recent years.In this thesis,the existence and multiplicity of solutions for two kinds of fractional Schr(?)dinger equations are explored by using variational methods.Firstly,this study probes into a class of fractional Schr(?)dinger equations containing nonlocal nonlinear terms.Known as one of fractional Choquard equations,it is characterized by the appearance of nonlocality of both fractional operators and nonlinear terms.By combining the Ekeland variational principle and the implicit function theorem,this equation is proved to have one least energy signchanging solution(has the lowest energy among all sign-changing solutions).The energy of such a solution is also proved to be between the ground state energy and twice that.Secondly,this study investigates one class of periodic fractional Schr(?)dinger equations,paying attention to the case of 0 in the spectral gap of fractional Schr(?)dinger operators.The main characteristic of this problem is that the corresponding energy functional is strongly indefinite.By means of the reduction method,this study proves the existence of the ground state solution,and further the existence of infinitely many geometrically distinct solutions in the case that the nonlinear term is odd.