Existence And Properties Of Solutions For Choquard Type Elliptic Equations

In this paper,we mainly study the existence of ground-state solutions for two kinds of Choquard elliptic equations with Hardy-Littlewood-Sobolev lower critical exponent and different Riesz potentials of nonlinear perturbation:(?)The research work in this paper is mainly divided into two-fold.Firstly,we discuss a special case of the weight function,that is,whether the corre-sponding Choquard equation(P)has a ground-state solution when the weight function q(x)is always equal to 1.We show that the functional J satisfies the mountain pass ge-ometry,when the weight function q(x)≡ 1.Then we give an estimation on the moun-tain pass energy level c,which is studied by Minimax Theorem.By using the Mountain Pass Theorem,we conclude that the Choquard equation(P)exists a ground-state so-lution,when N≥ 3,0<α<β,(N-2)<β<N,(N+β)/N<r<(N+β+2)N and λ>0.Secondly,we study the more general case of the weight function q(x).In this case,the energy functional Jq corresponding to the Choquard equation(P)still satisfies the mountain pass geometry,but its mountain pass energy level cannot be estimated by using the method of Chapter 2.Therefore,we prove that cq is attained by some u0 ∈Nq,when the weight function q(x)satisfies some assumptions in the Nehari manifold.And we conclude that the Choquard equation(P)has a ground-state so-lution when the weight function q(x)is a positive bounded continuous function,and N≥3,0<α<(N-1),α<β<N,(N+β)N<r<(N+β)(N-2),λ>0.