Some Topics In Bifurcations For Nonlinear Partial Differential Equations

This thesis is devoted to the investigations of bifurcation theory and its applications to nonlinear partial differential equations.We focus on two aspects.Firstly,we study global dynamic bifurcation theory of local semiflows and nonlinear evolution equations and its applications by means of dynamical system and the Conley index theory;Secondly,we study properties and the structure of positive solutions for coupled nonlinear Schr(?)dinger equations via variational methods and static bifurcation theory.For global dynamic bifurcation theory and its applications,in terms of invariantset bifurcation,we first establish two new global dynamic bifurcation theorems for local semiflows on complete metric spaces.These two theorems are effective tools when tackling dynamic bifurcation problems of nonlinear evolution equations as their conditions can be easily verified and are natural.It is worthwhile to note that one of these two theorems gives sufficient conditions for the global dynamic bifurcation branch to be unbounded.Then applications to nonlinear evolution equations are provided and a dynamical version of Rabinowitz’s global bifurcation theorem is established without assuming the condition of “crossing odd-multiplicity”.Moreover,for an important case of nonlinear evolution equations,we prove that the global dynamic bifurcation branch is unbounded.In addition,applying these theorems to an elliptic problem,some new results with global features are derived.Studies on coupled nonlinear Schr(?)dinger equations are divided into two parts.Firstly,we investigate the local and global bifurcation structure of positive solutions for a coupled nonlinear Schr(?)dinger system with the Neumann boundary condition.Our studies reveal that there is a richer structure of bifurcation phenomena for the Neumann system than for the counterpart of the system with the Dirichlet boundary condition.The reason is that the Neumann system has many synchronized solution branches,which also bring an essential difficulty for global bifurcation analysis.New techniques are needed to prove that the global bifurcation branches derived from different synchronized solution branches are independent of each other.We provide a detailed characterization of the global bifurcation structure of the Neumann system in one-dimensional space.Roughly speaking,our results demonstrate that for the Neumann system,there can be various “bifurcating tree” structure.However,for the Dirichlet system,there exists only one “bifurcating tree” structure.Secondly,we consider the uniqueness of positive solutions for some Schr(?)dinger equations with linear and nonlinear couplings by means of results on local and global bifurcation of the system.We prove some new results in several cases,especially in a degenerate case when the system is fully symmetric,improving relevant results in the literature.