Ambrosetti-Prodi Type Results For Some Nonlinear Differential Equations

This dissertation mainly establishes the Ambrosetti-Prodi type results for some nonlinear differential equations. Ambrosetti-Prodi type results describe the relation-ship between the number of the solutions for the problem and the parameter s. For example, we consider the following first-order periodic problem: If there exists a s1?K, such that the problem has no solution, at least one solution when s<s1,s>s1, respectively, then this result is called the Ambrosetti-Prodi type result for this problem.In this dissertation, firstly, by using the method of upper and lower solutions and topological degree techniques, we obtain the Ambrosetti-Prodi type results for the nonlinear second-order periodic boundary value problem and the mean curva-ture equation with the Neumann boundary condition in an annular domain. After that, we use the bifurcation theory to study the multiplicity results of nodal so-lutions for second-order Neumann boundary value problem. Then we estsblish a geometry explanation of the Ambrosetti-Prodi type results for second-order Neu-mann boundary value problem. The main works of the paper are as follows.1.For nonlinear second-order periodic boundary value problem where f:[0,T] × R2?R is continuous and uniformly in [0,T]. We obtain an Ambrosetti-Prodi type result for this problem where f doesn't need to satisfy the Bernstein-Nagumo condition. This result gives a complement for the corresponding result of Fabry, Mawhin and Nkashama [3]. 2.We study the mean curvature equation with the Neumann boundary condition in an annular domain where D={x?RN:R1?|x|<R2},1<R1<R2, f:[R1,R2]×R2?R is a continuous function. We transform this problem into where 1<R1<R2, f:[R1,R2]×R2?R is a continuous function,?:(-a,a)? R is an increasing homeomorphism and ?(0)=0,a>0. Then by using the method of the upper and lower solutions and topological degree techniques, we establish an Ambrosetti-Prodi type result for this problem. This result gives a complement for the corresponding result of Bereanu, Mawhin [11].3.By using the Rabinowitz bifuacation theory, we obtain the multiplicity results of nodal solutions for Then, for second-order Neumann boundary value problem with formal nonlinearity where f:R?R is continuous and C1(u+)p?f(u)?C2(u+)p,0<C1<C2, we use the Rabinowitz bifuacation theory to obtain the multiplicity results for this problem. Finally, we estsblish a geometry explanation of the Ambrosetti-Prodi type result for second-order Neumann boundary value problem.