| In this paper,the existence of infinity many solutions for a class of Hamiltonian system and two classes of elliptic systems are studied through variational methods and critical point theory.This dissertation is divided into three chapters,the main contents are as follows:In chapter 1,we consider a class of p-Laplace system:where p>1,T>0,F ∈ C([0,T],RN).We assume that F satisfies the following hypothe-ses:(H1)F(t,x)/|x|p→+∞ as |x|→∞ uniformly for a.e.t ∈[0,T];(H2)There exist constants a1,L1>0 such thatApplying the Fountain theorem of critical point theory,we get the main result as follow:Theorem 1.1.1 Suppose that F satisfies(H1)-(H3),then the system has infinity many periodic solutions.The result of chapter 1 is the generalization of the existing result in case of p=2.In chapter 2,we consider a class of nonlinear elliptic system with Neumann boundary condition:where Ω C RN(N ≥ 3)is a bounded with smooth boundary,(?)/(?)n is the outer normal derivative and f∈C(Ω×R,R),g∈C((?)Ω× R,R).We assume that f,g satisfies the following hypotheses:(F1)there exist fi∈L1(Ω;R+),gi∈L1((?)Ω;R+),α i ∈[0,1),i ＝ 1,2,such that and(F2)Applying the minimax of variational methods,we get the main result as follow:Theorem 2.1.1.Suppose that(F1)-(F2)hold,then,(i)there exists a sequence of solutions {un} of system which are critical points of functional φ,and φ(un)→+∞,as n→∞;(ii)there exists a sequence of solutions {um*} of system which are critical points of functional φ,and φ(um*)→-∞,as m→∞.The results of chapter 2 are to generalize the results of Hamiltonian system to elliptic system.In chapter 3,we consider a class of weighted nonlinear elliptic system:where 0∈Ω(?)RN(N≥ 3)is a bounded domain with smooth boundary,0 ≤ a<N-p/p,a≤b<a＋1,0 ≤μ<μa = dP,d= N-p-ap/p,λ>0,δ>0,1<q<p,and α,β>1 satisfyα+ β =p*,p*= pN/N-p(1+a-b)denotes the critical Sobolev exponent.By using the Caffarelli-Kohn-Nirenberg inequality and variational methods,we consider the system in N ehari manifold and get the main result as follow:Theorem 3.1.1 There exists A*>0 such that if λp/p-q+δp/(p-q)∈(0,A*),the system admits at least cat(Ω)+1 distinct positive solutions.The result of chapter 3 is the new result obtained by a class of weighted nonlinear elliptic system in N ehari manifold. |
