The Existence And Multiplicity Of Solutions Of The High-order Differential Equation Neumann Boundary Value Problems

This thesis is mainly composed of two chapters in which we discuss Neumann boundary value problem for ordinary differential equations. In Chapter 1, we study the existence of ground states for some 2m-order Neumann boundary value problems and new existence of ground states result are obtained by applying Minimax theory. In Chapter 2,new existence and multiplicity results are obtained for one kind of Neumann boundary value problem by the dual fountain theorem.In the following, we state the main results of this thesis concretely.In Chapters 1, we use the method to discuss the following 2mth-order ordinary differ-ential equation Neumann boundary value problem, where f∈C([0,1] x R1), and obtain the following theorems.Theorem 1.3.4. If fsatisfies the following conditions.(f1) there exists C0> 0 such that|f(t,u)|≤C0(｜u｜+｜u｜p-1), (t,u)∈[0,1] x R1, where p>2;(f2) f(t, u)= o(u), uâ†'0, uniformly for t∈[0,1];(f3) there exists a> 2 such that aF(t,u)≤uf(t,u), (t,u)∈[0,1] x R1;(f4) there exists R> 0 such that (?) F(t, u)> 0;(f5) for all t∈[0,1], f(t,u)/｜u｜is strictly increasing in u, then the problem (1.1.1) has a ground state in C2m[0,1].In Chapter 2,we discuss the following 2mth-order ordinary differential equation Neu-mann boundary value problem. whereλ,μis parameters.The following theorem are the main results of this Chapter.Theorem 2.3.1. If f(t, v)=μ｜v｜q-2v+λ｜v｜p-2v, then for everyμ>0,λ∈R1,problem (2.1.1) has a sequence of solutions{vn}(?) such thatψ(vn)< 0 andψ(vn)â†'> 0, as nâ†'∞ Theorems 1.3.4 and 2.3.1 are new results about the existence and multiplicity of so-lutions to the 2mth-order ordinary differential equation Neumann boundary value prob-lem, this is new in this paper.