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 ?(?)RN is a bounded open set with smooth boundary in ?.N>3,0<?<N,1<q<2,2?*= N-2/2N-?,and ?>0 is parameter.There are two continuous weight functions,f?,g satisfying the following conditions(f2)There exist two positive constants ?0,?0 such that B(0,2?o)(?)? and f?(x)??0;for every x ? B(0,2?0);(g2)|g+|? = g(0)= maxx??g(x),(?)x ? B(0,2?0),g(x)>0,there exists?(?>N-?),such that g(x)-g(0)= 0(|x|?),(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 ?(?)RN is a bounded open set with smooth boundary in RN(N ? 3),with C2 boundary(?)?,0<?<N,2?*=N-2/2N-?,and ?>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 ??,t ? 0.Let u±= max{±u,0},F(x,t)=(?)f(x,?)d?.(f1)f ? C(?×R+,R),limt?0+t/f(x,t)=+ ? and limt?? t22*-1/f(x,t)= 0 uniformly for x ? ?;(f2)? × R+ ? R is nondecreasing with respect to the second variable.