The Solutions To A Mixed Coupled Schr(?)dinger System With Quadratic Self-focusing

Schr(?)dinger systems are widely used in the fields of classical mechanics and quantum physics.The self-focusing term has an important impact on the energy band structure and quantum state evolution of mixed coupled Schr(?)dinger systems.In this thesis,we study the positive solutions,ground state solutions,sign changing solutions and normalized solutions to a mixed coupled Schr(?)dinger system with quadratic self-focusing The behavior of solutions may be different when parameters vary in different ranges.In the system(Sqm),the quadratic self-focusing term and the cubic coupling term are not homogeneous,and the quadratic coupling term is partially symmetric,which bring some difficulties to the study of the problem.In this thesis,we overcome these difficulties by using the properties of energy,some estimates,constructing auxiliary operators,and rearrangements.The full text consists of six chapters.The first and second chapters mainly describe the background of the research problem and the research status of Schr(?)dinger systems at home and abroad,summarize the main results of the research problem,and introduce the preliminary knowledge needed in this thesis.The main content of this thesis lies from the third to fifth chapter.In Chapter 3,we study the existence of positive solutions and ground state solutions to the system(Sqm)by the variational method.When N≤3,the parameters λ1,λ2,μ1,μ2 are positive and ρ=0,we improve an existed result when β is positive,that is obtaining the nonexistence result of the positive solution if either the auxiliary matrix A is positive definite and λ2≥λ1 or A is negative definite and λ1≥λ2.In addition,whenρ is positive,either λ2>(μ2/μ1)2/(4-N)or there exists a positive number β0 such thatβ≥β0,the system(Sqm)has a ground state solution.Furthermore,if the parameter βis positive,then the system(Sqm)has a nontrivial positive radial ground state solution.When N≥6 and β is positive,we prove that the system(Sqm)has no positive solution.Chapter 4 studies the existence of sign-changing solutions to the system(Sqm)by constructing invariant sets containing semi-trivial and signed solutions of the system and finding solutions outside these invariant sets.When λ1,λ2,μ1,μ2 are all positive,β,ρare both negative and |β| is sufficiently small,the existence of sign-changing solutions is proved by constructing two auxiliary operators and using an abstract theorem based on the invariant set of descending flow.Chapter 5 investigates the existence of solutions to the system(Sqm)under the normalized mass conditions ∫RNu2=a12 and ∫RN v2=a22.When the parameters λ1 and λ2 are unknown,μ1,μ2 and β are positive,we discuss the existence of normalized solutions in different spatial dimensions in two cases,that is ρ=0 and p>0.When N=1,2,3 andρ=0,the energy functional is always bounded from below.At this point,the existence of normalized ground state solutions can be proved by using rearrangement.When N=1 and p>0,it can be similarly obtain the existence of the ground state normalized solution.When N=2 and p>0,the behavior of the energy functional changes and is no longer always bounded from below.When the parameter p belongs to(0,|U0|22/Λ),the existence of a positive normalized solution can be obtained,where U0 is the unique positive radial solution to the equation-Δu+u=u3 and A=max{a12,a22}.Chapter 6 summarizes the research contents and provides an outlook for future research.