Existence And Multiplicity Of Solutions For Two Classes Of Nonlinear Differential Equations With Nonlinear Boundary Conditions

In this thesis, we study existence and multiplicity of solutions of one dimensional and high dimensional for given curvature equations with the nonlinear boundary con-ditions (minimum and maximum) in Minkowski space by using Leray-Schauder theory and Borsuk theorem, respectively. Using fixed point index theory, we also consider existence of positive solutions for an elastic beam equation with nonlinear boundary conditions. We describe them in detail as follows.1. We study multiplicity solutions of the singular φ-Laplacian equation with mini-mum and maximum conditions where φ:(-α, α) → R(0< α< ∞) is an odd increasing homeomorphism, F C1[0,T]→L1[0,T] is a unbounded operator, T> 1 is a constant and A, B ∈ R are satisfying B> A. We give the sufficient conditions of two different solutions of the problem (P1) with the nonlinearity satisfying unbounded conditions. The main results partly extend the corresponding results of Bereanu and Mawhin [J. Differential Equations,2007], Bereanu and Mawhin [J. Math. Anal. Appl.,2009]. When φ= I, the main results not only directly improve Brykalov [Comm. Appl. Nonlinear Anal.,1996] and Stanek [Math. Nachr.,1998], but also partly improve the main results of Stanek [Nonlinear Anal.,1997]. Finally, we give an example to illustrate our main results.2. We study multiplicity of radial solutions for the quasilinear problems with min- imum and maximum in Minkowski space where φn(z) = (?), z ∈ RN, R1, R2, A, B ∈ R are constants and satisfying 1 < R1 < R2-1, A < B; |·| denote the Euclidean norm in RN, F : C1[R1,R2] → L1[R1,R2] is an unbounded operator. The two different radial solutions of the problem (P2) are new. The main results of the high dimension are different from the one dimension’s results, which is only right in special annulus. At last, we give a simple example to apply our main results.3. By using the fixed point index theory, we consider existence of positive solutions for an elastic beam equation with nonlinear boundary conditions where λ > 0,μ > 0 are parameters, f : [0,1] × [0, +∞) → (0, +∞) is continuous. We obtain that problem (P3) has at least one positive solution with the nonlinearity f satisfying superlinear growth condition at infinity. The results improve the correspond-ing results for the only one parameter in Liu Yang, Chen Heibo and Yang XiaoXia [Appl. Math. Lett., 2011], Li Shunyong and Zhao Xiaoqin [Appl. Math. Comp., 2012], Cabada and Tersian [Appl. Math. Comp., 2013]. In the end, we give a simple example to describe the existence and nonexistence of positive solutions with two parameters varying.