The Existence And Multiplicity Of Solutions For Several Classes Of Elliptic Problems

In this thesis,by using the variational method,Nehari method and some analysis techniques,we study the existence and multiplicity of solutions for a class of elliptic boundary value problem and some elliptic equations with nonlocal terms.Firstly,we consider the following semilinear elliptic problem where Ω(?)RN(N ≥ 3)is an open bounded domain with smooth boundary(?)Ω,a∈LN/2(Ω),and the nonlinearity f∈C(Ω×R,R)satisfies some more gener-al subcritical conditions.We get that problem(0.5)possesses one or infinitely multiple solutions for f with different conditions.Secondly,we investigate the existence of the following class of Choquard equation where N∈N,N≥3,α∈(0,N),the function Iα:RN\{0}→R is the Riesz potential,λ>0 is a parameter,p=N+α/N-2 is the upper Hardy-Littlewood-Sobolev critical exponent and q∈(2,2*).We prove that there exists λ0>0 such that forλ≥λ0,problem(0.6)possesses one positive radial solution.Next,we study the following zero mass Choquard equation(f2)there exists t0∈R\{0} such that F(t0)≠ 0.We reach the conclusion that(0.7)possesses one nontrivial solution by con-structing a Pohozaev-Palais-Smale sequence.Finally,we investigate the existence of one positive ground state solution for the following class of Choquard equation where N∈N,N≥3,α∈(O,N),Iα is the Riesz potential,V is asymptotically periodic.We prove that(0.8)possesses one positive ground state solution,and if V is periodic,then the set of ground state solutions of(0.8)is compact(up to a translation).