Solutions Of The 2mth-order Ordinary Differential Equation Dirichlet Boundary Value Problems

Ordinary differential equation Dirichlet boundary value problems are more general ones in boundary value problems. Many researchers studied these problems by employing the topological degree theory and fixed point index theory (see [16]-[23]) or by using Morse theory (see [24]-[31]) or the critical point theory (see [3], [5], [6], [7], [9]-[15]), and obtained the existence and multiplicity of solutions. It is necessary to list these papers ([3], [5], [6], [8], [9]) which are helpful to this paper.In paper [5], the author used the new critical point theory established in paper [3] to discuss the fourth-order boundary value problem and obtained the results of four solutions and six solutions. Motivated by these papers and their results, in Chapter one, we apply the method to the following 4mth-order ordinary differential equation Dirichlet boundary value problem: where using the method in paper [6] which transformed boundary value problem into the integral equation, and obtain the following theorems:(H1) there exist a strict subsolution a and a strict supersolutionβof problem (1.1.1), a<β, and a,βall satisfy the condition of boundary value of (1.1.1);(H2) f(t, u) is strictly increasing in u;(H3) f(t,u) is Lipschitz continuous in u;(H4) there existμ> 2 and M> 0 such that where then the problem (1.1.1) have at least four solutions.(H5)α1<β1,α2<β2 are two pairs of strict subsolutions and strict supersolutions of problem (1.1.1) and they all satisfy the condition of boundary value of (1.1.1), then the problem (1.1.1) has at least six solutions.In [8, Theorem 3.20,4.2, p.65-73], the author used minimax theory to discuss a kind of second order elliptic equation and obtained the existence of ground states solution and infinite multiplicity of solutions. In chapters two and three, we use the method to discuss the following 2mth-order ordinary differential equation Dirichlet boundary value problem: where and obtain the following theorems:(H6) there exists Co> 0 such that where p>2; (H7) f(t,u)= o(u),u→0, uniformly for t∈[0,1]; (H8) there exists a> 2 such that (H9) there exists R> 0 such that (Hio) for all is strictly increasing in u, then the problem (2.1.1) has a ground state in C2m[0,1].Theorems 2.3.4 and 3.3.2 are new results about the existence and multiplicity of so-lutions to the 2mth-order ordinary differential equation Dirichlet boundary value problem, this is new in this paper.