Global Structure Of Positive Solutions For The One-dimensional P-laplacian Problems

In this thesis,by using the Global bifurcation theorem,we study the existence and multiplicity of two classes of differential boundary value problems involving one-dimensional p-Laplacian operator.And by using the Time-map analysis method,we establish the Ambrosetti-Prodi type results for a class of semipositone Neumann problem.The main works of the paper are as follows.1.Using the global bifurcation theory,we study the Dirichlet boundary value problem with one-dimensional p-Laplacian operatorWe show the S-shaped connected component in the solutions set under the assump-tions f0 = ? and f?=0.where,?p(s)=|s|p-2,p>1,?>0 is a parame-ter,h ? C([0,1],(0,?)),f ? C[0,?),f(0)= 0,f(s)>0,for all s>0.The main results ont only generalize the corresponding results with ? ? 1 in Y.Lee,S.Kim and E.Lee[Abstr.Appl.Anal.,2014],but also complete the condition 0<f0<? in Y.Lee and I.Sim[J.Differential Equations.,2006].2.When function h with a sign-changing weight and symmetric on[0,1],by us-ing global bifurcation theorem,we study the one-dimensional p-Laplacian boundary value problemWe show the global structure of symmetric positive solutions in the solutions set under the assumptions f0 = ? and f?=0.Where,?p(s)= |s|p-2s,p>1,?>0 is a parameter,f ? C[0,?),f(0)= 0,f(s)>0,for all s>0.The results of this Chapter partially improve the main ones of I.Sim and S.Tana-ka[Appl.Math.Lett.,2015],and overcome the difficulties of estimate of ||u||.If? ? 1 and h>0,this result gives a complement for the corresponding result of H.Feng,H.Pang and W.Ge[Nonlinear Anal.,2008].3.Using time-map analysis,we establishes the Ambrosetti-Prodi type results for a Neumann problemIf there exists a t0 ?R,such that the problem has no solution,at least one solution and at least three solutions when t<t0,t = t0 and t>to,respectively,then this result is called the Ambrosetti-Prodi type results for this problem.And at last,we obtain the exact number of positive solutions of the above problem with appropriate conditions.Where ?p,(s)= |s|p-2s,p>1,t is a parameter,f ?C2[0,?),f'(u)>0 for all u>0.