The Solutions Of A Class Of Elliptic Boundary Value Problems And Hamiltonian System

With the continuous development of science and technology, all kinds of nonlinear problem have aroused people’s wide attention. Nonlinear partial differential equations stem from applied mathematics, physics, control theory and other applied sciences, they are the most active research topics in the field of nonlinear science, the elliptic boundary value problems and Hamiltonian system problems are the hot topics these years, and they are also quite important research fields of the partial differential research.With further research, under more complicated or weaker conditions, we investi-gated the existence of solutions of several types of differential equations.In this paper, some new results of the existence of nontrivial solutions for a class of differential equations have been obtained by The mountain pass theorem.This paper is divided into three chapters as follows:Chapter1Preliminary knowledge, we introduce some definitions and some im-portant the theorems which can be used during proof.Chapter2In Chapter2, we are interested in existence of solutions of elliptic boundary value problems with mixed type nonlinearities where Ω (?) RN(N>2) is a bounded open domain with smooth boundary (?)Q and f∈C(Ω×R,R). satisfy(F1) F(x, u)=-K(x, u)+W(x, u), K, W: Ω x R1→R1are C1-maps.(K1) There are two positive constants b1and b2such that b1|u|2≤K(x,u)≤b2|u|2, for all (x,u)∈Ω x R1.(K2) There exists (?)∈(1,2] such that K(x,u)≤Ku(x,u)u≤(?)K(x,u), for all (x,u)∈Ω x R1.W(x,u)≥0and Wu(x,u)=o(|u|) as|u|→0uniformly in x. (W2) W(x,u)/u2→∞as|u|→∞uniformly in x.(W3)Set W(x,u):=1/2Wu(x,u)u-W(x,u),W(x,u)>0if u≠0,W(x,u)→∞as|u|→∞uniformly in x,and there exist r0>0and σ>max{1,N/2}such that|Wu(x,u)|σ≤c0W(x,u)|u|σif|u|≥r0.By verifying the mountain pass structure and(C)condition,we obtain the exis-tence of nontrivial solutions of the elliptic boundary value problems(P)).This chapter is divided into three sections. The first section is an introduction,mainly introduce some previous results of elliptic boundary value problems.The second section are some preliminary knowledge.In third section,we prove the existence of nontrivial solutions of the elliptic boundary value problem.Chapter3In Chapter3,we are interested in the existence of periodic solution for a second-order asymptotically linear Hamililtonian system where T>0. Let Assume that(F1):F(t,x)=-K(t,x)+W(t,x),where K,W:[0,T]×RR→R are C1-maps；(K1):there exist γ∈(1,2],α>0such that K(t,x)≥α|x|γ,for all(t,x)∈[0,T]×RN；(K2):there exist (?)∈(1,2]suchh thatfor all(t,x)∈[0,T]×RN；(W1):▽W(t,x)=o(|x|)as|x|→0uniformly with respect to t；(W2):|▽W(t,x)-V∞(t)x|/|x|→0as|x|→∞uniformly with respect to t,where V∞∈L∞([0,t],R)with inf[0,T]V∞>2(M0+1)/k,k=1/(1+ω2),ω=2π/T；(W3):W(t,x)→+∞as|x|→∞.This chapter is divided into three sections. The first section is an introduction, mainly introduce some previous results of Hamililtonian systems problems.The second section are some preliminary knowledge.In third section,we prove the existence of nontrivial periodic solutions of the Hamililtonian systems problem(HS).