The Bifurcation Method For The Positive Solutions Of A Class Of Nonlinear Differential Equations

This thesis of Master is composed of five chapters, which mainly studies a class of nonlinear differential equations, when the nonlinear part of the equations has some special conditions, using nonlinear analysis and bifurcation theorem we can know how many solutions the nonlinear differential equations have if theλchanges. Then we continue to research on this nonlinear differential equations more furtherly, when this positive solution is a degenerate critical solution, we can infer an integal formula E(α) firstly and it can judge the solutions curve how to change it's direction in the (λ, u) plane by it's values.In the first chapters, the thesis mainly introduces a class of nonlinear differential equations of the history developement and the recent development of the research in this equations.In the second chapters, it mainly introduces us some preliminary knowledge and related lemma about studying this class of the nonlinear differential equations.In the third chapter, it mainly studies when the nonlinear part of the equations has some special conditions, using nonlinear analysis and bifurcation theorem we can know how many solutions the nonlinear differential equations have if theλchanges.In the fourth chapter, it continue to research on the nonlinear differential equations more furtherly, when this positive solution is a degenerate critical solution, we can infer an integal formula E(α) firstly and it can judge the solutions curve how to change it's direction in the (λ, u) plane by it's values.In the fifth chapter, it mainly conclues the thesis and there are many open questions which need to solve.