Two Kinds Of Solutions To The Choquard Equation With A Critical Exponent

The main goal of this thesis is to consider the multiplicity of positive solution for the nonlinear Choquard equation.In the case one,we study a nonlinear Choquard system with weighted terms and critical Hardy-Littlewood-Sobolev exponent.We apply variational methods and Ljusternik-Schnirelmann category to prove the multiple positive solutions for this problem.where Ω(?)RN is a bounded open set with smooth boundary in Ω.N>3,0<μ<N,1<q<2,2μ*= N-2/2N-μ,and λ>0 is parameter.There are two continuous weight functions,fλ,g satisfying the following conditions(f2)There exist two positive constants β0,ρ0 such that B(0,2ρo)(?)Ω and fλ(x)≥β0;for every x ∈ B(0,2ρ0);(g2)|g+|∞ = g(0)= maxx∈Ωg(x),(?)x ∈ B(0,2ρ0),g(x)>0,there existsβ(β>N-μ),such that g(x)-g(0)= 0(|x|β),(x → 0).In the case two,we study a existence and multiplicity of solutions for a class of Choquard equations with Hardy-Littewood-Sobolev critical exponent.The existence of positive solu-tion for this problem is established by Mountain Pass Theorem. where Ω(?)RN is a bounded open set with smooth boundary in RN(N ≥ 3),with C2 boundary(?)Ω,0<μ<N,2μ*=N-2/2N-μ,and λ>0 is parameter,it is obvious that the value of f(x,t)for t<0 are irrelevant,so we may define f(x,t)= 0 for x ∈Ω,t ≤ 0.Let u±= max{±u,0},F(x,t)=(?)f(x,τ)dτ.(f1)f ∈ C(Ω×R+,R),limt→0+t/f(x,t)=+ ∞ and limt→∞ t22*-1/f(x,t)= 0 uniformly for x ∈ Ω;(f2)Ω × R+ → R is nondecreasing with respect to the second variable.