Existence And Multiplicity Of Solutions For The Critical Choquard Equation

In this thesis,we study the following critical Choquard equations:homogeneous and nonhomogeneous critical Choquard equation on a bounded domain,strongly indefinite critical Choquard equation,critical Choquard equation with deepening potential well,and prove the existence,multiplicity and regularity of solution by variational methods.In Chapter 1,we introduce the research backgrounds and recent developments of Choquard equation.Then we recall some preliminary knowledge and state the main results obtained in the thesis.In Chapter 2,we investigate the existence of the following Brezis-Nirenberg type critical problem for Choquard equation (?) where ? is a smooth bounded domain of RN,? is a real parameter,N?3,0<?<N,2?*=(2N-?)/(N-2)is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.We prove the existence and nonexistence of solutions for the above problem by variational methods.In Chapter 3,we establish the existence and multiplicity of solution for the following critical Choquard equation under the perturbation of both sublinear and suplinear terms (?) where 0<q<1,1<p<2*-1.In Chapter 4,we are going to consider the existence and multiplicity of solution for the following critical nonhomogeneous Choquard equation (?) Here ? is a smooth bounded domain of RN,N ?7,0<?<N,0 in interior of ?,0<?<?1,where ?1 is the first eigenvalue of-?,f(x)? L?(?)and f(x)? 0.In Chapter 5,we investigate the existence of the following strongly indefinite critical Choquard equation (?) where N?4,0<?<4 and G(x,u)=(?).We assume that 0 lies in a gap of the spectrum of-? + V and g(x,u)is of critical growth.In Chapter 6,we investigate the existence and multiplicity of solution for following critical Choquard equation with deepening potential well (?) If ?>0 is a constant such that the operator-?+ ?V(x)-? is non-degenerate,we prove the existence of ground state solutions for ? large enough and also characterize the asymptotic behavior of the solutions as the parameter ? goes to infinity.Furthermore,there exists ?*?(0,?1)such that for any ??(0,?*),we obtain the existence of multiple solutions by the Lusternik-Schnirelmann category theory.In Chapter 7,we give some problems for further exploration.