Positive Solutions,Negative Solutions And Sign-changing Solutions For Several Schrodinger Equations

In recent years,the research on the solution of fractional Schr(?)dinger equation has attracted wide attention both in mathematics and in practice,and its theory has been widely used to describe the micro particle state,quantum mechanics,molecular spectrum and other fields.It has a far-reaching impact on many other disciplines such as mathematical physics and Biomathematics.In this paper,The existence of solutions for two kinds of fractional Schr(?)dinger equations is studied by using the variational method and some tools and analytical methods in the critical point theory.Details as follows:The first one,we study this form of fractional Schr(?)dinger equation:and where (?)?(0,1),V(x)?C(R~3,R),f?C(R,R),in the whole three-dimension.Using the non-Nehari manifold under some specfic conditions,we can get at least one minimum energy sign-solution to the problem.Moreover,by using the deformation lemma and topological degree theory,we can obtain at least one minimum energy sign-changing solution for a class of fractional Schr(?)dinger-Poisson equations related to this problem.The second one,we study the existence of solutions for the following fractional Schr(?)dinger equation:where (?)?(0,1),V(x)?C(R~3,R),f?C(R?R) is Carath(?)odory function,satisfying the subcritical condition.Using the mountain lemma,we can obtain a positive solution and a negative solution to the problem.In addition,using critical point theory,we can get infinitely many sign-changing solutions of the problem.There are three chapters in this paper.The first chapter is the introduction,which mainly introduces the research background and preliminary knowledge of the fractional Schr(?)dinger equation.The second chapter uses the non-Nehari method and topological degree theory study the existence of the sign-changing solution of the second fractional Schr(?)dinger equation.The third chapter studies the existence of the positive solution,negative solution and infinitely many sign-changing solu-tions of the second fractional Schr(?)dinger equation.