The Existence And Multiplicity Of Solutions To Schr(?)dinger Equations

In recent years,there have been many important achievements on the Laplace equation and the multiple positive solutions,ground state solutions,and sign change solutions of the equation system.In addition,coupled Schrodinger equations including Laplace operators have gradually appeared in several branches of physics.It is more common that they can be used to describe electromagnetic field problems in nonlinear Schrodinger field.Therefore,the existence of the normal ground state solution of the Schrodinger equations has been widely studied,especially the ground state solution of the critical nonlinear growth Schrodinger equations.But the classical Brezis and Nirenberg methods do not seem to be applicable to all situations,so this problem still need further research.Due to the importance of Schrodinger equation to mathematics and other disciplines,as well as the research status of Schrodinger equation so far,this article will focus on the existence of solutions to several types of Schrodinger equations with different characteristics.First,use the variational method to show that the fractional Schrodinger equation with interference term has at least two solutions.On the one hand,it is proved that the Nehari manifold corresponding to the equation is not empty,and its corresponding minimum is uniformly bounded.On the other hand,it is proved that the minimum value can be reached and the equation has two different solutions.Secondly,the existence of the ground state solution of the fractional Schrodinger system with nonlinear critical exponents is discussed with the help of the Mountain Road Theorem.To facilitate the study,we first consider the existence of the ground state solution of the system in the limit case(?=RN and ?=0).Using a method similar to the limit case,we get the relevant conclusions about the existence of the ground state solutions.Finally,the ground state solutions of the above system in the magnetic environment are discussed separately.By discussing the properties of the functions related to the energy functional of the system,it is proved that the system must have a ground state solution when N>4s.According to the value range of the parameter?,whenN=4s,the existence of semi-trivial pairs corresponding to the Nehari manifold is discussed and specific proofs are given.