Existence Of Ground State Solutions For Choquard Equations With Critical Term

In this thesis,by using the variational method and some analysis techniques,we consider the following Choquard equation where N ? N,N ? 3,??(0,N),I?:RN\{0} ? R is the Riesz potential of order?,F(t)=?0t f(s)ds and f,g? C(R,R).Firstly,we study problem(0.0.1)with the critical Sobolev perturbation term,namely where 2*=2N/N-2is called the critical Sobolev exponent.If f satisfies the subcritical Hardy-Littlewood-Sobolev growth conditions,we prove that problem(0.0.2)has one positive ground state solution by constructing Pohozaev-Palais-Smale sequence from minimax theorem and the local compactness condition.Secondly,we research problem(0.0.1)with the upper critical growth term and general perturbation term,that is where 2*?-2(N+?)/N-2(2?*/2= is called the upper critical Hardy-Littlewood-Sobolev expo-nent).If g satisfies the subcritical Sobolev growth conditions,we use Lions lemma to overcome the lack of compactness,and then apply minimax theorem to obtain that problem(0.0.3)has at least one positive radial ground state solution.As a final topic,we consider problem(0.0.1)with the upper critical Hardy-Littlewood-Sobolev growth general term,that isIf f satisfies the general critical growth conditions,we prove that problem(0.0.4)has at least one positive ground state solution by using Pohozaev method.