| This paper aims to study two kinds of nonlocal problems:one is the critical Hartree equation with axisymmetric potentials,and the other is critical Hartree equation on a bounded domain.We study the existence and asymptotic behavior of solutions of two kinds of critical Hartree equations by using the finite-dimensional reduction method,local Pohozaev identities and blow-up analysis.In chapter 1,we introduce the research backgrounds and recent developments of nonlinear Hartree equation.Then we recall some preliminary knowledge and state the main results obtained in the thesis.In Chapter 2,we investigate a class of critical Hartree equations with axisymmetric potentials where(x’,x")∈ R2×R4,V(|x’|,x")is a bounded nonnegative function in R+×R4,and*stands for the standard convolution.The equation is critical in the sense of the HardyLittlewood-Sobolev inequality.We first prove a nondegeneracy result for the critical Hartree equation in RN.Secondly,by applying a finite-dimensional reduction argument and developing novel local Pohozaev identities,we prove that if the function r2V(r,x")has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.In chapter 3,we investigate the critical Hartree equations on bounded domain where N≥4,0<μ≤4,ε>0 is a small parameter,Ω is a smooth bounded open domain in RN,and 2μ*=(2N-μ)/(N-2)is the critical exponent in the sense of the Hardy-LittlewoodSobolev inequality.We first establish a nonlocal version of global compactness lemma.Secondly,by using finite-dimensional reduction method,we prove that,as ε→0,the solution uε of the problem blows up exactly at a critical point of the Robin function that cannot be on the boundary of Ω.In Chapter 4,we investigate the critical Hartree equations on bounded domain where N≥5,μ∈(0,4],Ω is a bounded smooth domain in RN,and 2μ*=(2N-μ)/(N-2)is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.By applying the reduction arguments,we prove that the above problem has a family of solutions uε concentrating around the critical point of Robin function under some suitable assumptions,if ε→0,N≥5,μ∈(0,4]sufficiently close to 0,4 or=4.In Chapter 5,we investigate the critical Hartree equations on bounded domain where N≥4,0<μ≤4,ε>0 is a small parameter,Ω is a bounded domain in RN,and 2μ*=(2N-μ)/(N-2)is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.By establishing various versions of local Pohozaev identities and applying blow-up analysis,we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation.Next we prove the local uniqueness of the blowup solutions that concentrates at the non-degenerate critical point of the Robin function for ε small.In Chapter 6,we give a conclusion and outlook. |
