Fourth-order Neumann Boundary Value Problem

In this paper, we discuss the existence of two sign-changing solutions to the fol-lowing nonlinear fourth-order Neumann boundary value problem:This thesis is mainly composed of three chapters: Chapter 1 is introuduction. InChapter 2, we introuduce some preliminaries and some lemmas, and using the fixedpointed theory, critical point theory and morse theory, we prove that the equation (1.1)has one positive solution and one sign-changing solution at least in certain conditions.Furthermore, we calculate the fixed pointed index of the first sign-changing solution bycritical group, this conclusion cannot be obtained using the traditional fixed pointedtheory and critical point theory. In Chapter 3, and using the fixed pointed theory, weprove that the equation (1.1) has one positive solution and two sign-changing solutionsat least in certain conditions by simple calculation.Recently, many authors discuss the equation:They get many faily good conclusions, for instance[5, 9, 12, 13], but people seldomdiscuss the equation(1.1). The main cause has three aspects:(1)if using traditionalfixed pointed theory, we cannot contain the essential conclusions differnt from equa-tion(1.1'),(2)we convert the BVP(1.1) into the functional of H₀²[0,1] using traditionalcritical point theory is very difficult. (3)Not notice the detailed quality of isolatedcritical point.In this paper,We can solve these problems of being listed by using theMorse theory and the character of K².The main difference from the ordinary literatures has two aspects: (1) the non-linearity of equation(1.1) is jumping at∞,so we cannot prove the p.s condition usingordinary proof.(2) we prove the conclusion of literature [4] on the weaker conditions,andcontain the sencond sign-changing solution.The detailed assumptions on f are listed below:(f₁)f∈C¹([0,1]×R,R), and f is increasing on R. (f₂) f(t,0)=0 for all t∈[0,1] andλ_k＜f_u^′(t,0)＜λ_k+1 for some k≥2.(f₃) lim sup_{u→+∞}f(t,u)/u＜λ₁ uniformly in t∈[0,1].(f₄)λ₁＜lim inf_{u→-∞}f(t,u)/u≤lim sup_{u→-∞}f(t,u)/u＜+∞uniformly int∈[0,1].Whereλ_k is the eigenvalue of equation(1.1),λ_k＞0,k∈NThe following theorem is the main result of this paper.Theorem 1.1. Suppose that (f₁)-(f₄) hold. Then the BVP (1.1) has one positivesolution and two sign-changing solutions.