Existence And Quantitative Properties Of Solutions To Nonautonomous Choquard-Type Equations

In recent years,Choquard-type equations have been widely used in quantum mechanics,nonlinear optics and Bose-Einstein condensates.In this thesis,we study the existence and quantitative properties of solutions to the following Choquard-type equation-Δu+（V+λ）u=[Q*（KF（u））]Kf（u）in R^N,（1）where N≥)3,V and K are potential functions,Q=I_α+μI_β,μ≥0 is a constant,Ia and I_β are the Riesz potentials,α,β ∈（0,N）,that is,（?） We study the ground state solution of（1）when λ is a fixed constant,and study the ground state normalized solution of（1）when the mass of λ is fixed and λ appears as a Lagrange multiplier.This thesis contains seven chapters.In the first chapter,we introduce some research background of Choquard-type equations and the research advance at home and abroad,and state the main research results of this thesis.In Chapter 2,we give the notations and preparations used in this article.Our main research contents are in Chapters 3-6.In Chapter 3,we study the existence of ground state solutions to the Choquard-type equation（1）,where λ=0,Q=I_α,α∈（（N-4）+,N）,F（s）=|s|^p,f（s）=|s|^p-2s,s ∈R,p ∈（2,2_α*）,2_α*:=（N+α）/（N-2）is the Hardy-Littlewood-Sobolev upper critical exponent,V=a+vV₀,K = b+K₀,a,b are positive constants,v is a nonnegative constant,V₀,K₀∈C（R^N,R+）\{0} are exponential decay.When V₀ decays faster than K₀,for any given v,the equation（1）has a positive ground state solution,that is Theorem 3.4.2.When V₀ decays slower than K₀,there exists v*∈（0,∞）such that the equation（1）admits a positive ground state solution for v∈[0,v*),while v ∈（v*,∞）,the equation（1）has no ground state solutions,that is Theorem 3.4.5.In Chapter 3,non-critical Choquard-type equations are considered,and in Chapter 4,we study doubly critical growth cases.From Chapter 4 onwards,K=1.Specifically,in Chapter 4,we investigate the existence and boundedness of ground state solutions to the Choquard-type equation（1）,where N≥5,λ=0,Q=I_α,α∈（0,N-4）,f（s）=|s|^p-2s+|s|q-2s,s ∈ R,F（s）=∫0sf（t）dt,p=2α:=（N+α）/N is the Hardy-Littlewood-Sobolev lower critical exponent,q= 2*,V ∈ C（R^N）satisfies certain assumptions.Through the Nehari manifold method,the existence of the positive ground state solution of equation（1）is proved,that is Theorem 4.3.2.Through truncation techniques and Moser iterations,it is proved that the positive solution of equation（1）is bounded,that is Theorem 4.4.3.In Chapters 3 and 4,the unconstrained problem is studied.In Chapters 5 and 6,the constraint problem with the constraint condition ∫R^N u2=a>0 is studied,and λ appears as a Lagrange multiplier,which is more mathematically complex and challenging than the unconstrained problem.In Chapter 5,the mass subcritical case is examined.In Chapter 6,the mass supercritical case is studied,especially the Hardy-Littlewood-Sobolev critical case is considered by the subcritical approximation method.Specifically,in Chapter 5,the existence of ground state normalized solutions to the Choquard-type equation（1）is studied,where Q=I_α,α∈（0,N）,V ∈C（R^N）and f ∈ C（R）satisfy certain assumptions,F（s）=f₀^s f（t）dt.Through the minimization argument,it is proved that there exists a₀ ∈[0,∞)such that the equation（1）has a positive ground state normalized solution when a E（a₀,∞）.If a₀>0,then the equation（1）has no ground state normalized solution when a E（0,a₀）,that is Theorem 5.4.8.If V=0,then the condition of f in Theorem 5.4.8 can be weakened appropriately,that is Theorem 5.3.4.In Chapter 6,the existence of ground state normalized solutions to the Choquardtype equation（1）is investigated,where Q=I_α+μI_β,μ>0,α and β satisfy 0<α≤β<min{（Na+4）/（N-2）,2α/2_α},F（s）=|s|,f（s）=|s|^p-2s,s∈R,p ∈(β,2_α*],β:=（N+β+2）/N is the mass critical exponent,V E C1（R^N）satisfies certain assumptions.By Nehari-Pohozave manifold and subcritical approximation method,it is proved that if p ∈（β,2_α*）,then for any given μ,equation（1）has a positive ground state normalized solution with λ>0;If α≠β and p=2_α*,then there exists μa>0 such that the equation（1）has a positive ground state normalized solution with λ>0 when μ≥μa,that is Theorem 6.5.3.We also consider in this chapter the regularity and Pohozave identity of solutions to a more general Choquardtype equation using truncation techniques and Moser iterations,that is Theorem 6.1.5.Finally,we give some summary about the study of the Choquard-type equation（1）,as well as possible research directions in the future.