The Existence And Exact Number Of Positive Solutions For The Prescribed Mean Curvature Problems In Minkowski Space

The degree thesis,the exact number of positive solutions of Dirichlet boundary value problems for the prescribed mean curvature equation is obtained by using time map method,the existence and exact number of multiple positive solutions of Neumann boundary value problem for the prescribed mean curvature equation are obtained by using time map method and bifurcation theory.In the first chapter of the thesis,we mainly introduce the background,research s-tatus,main work preliminary knowledge of Dirichlet boundary value problem and Neu-mann boundary value problem for the prescribed mean curvature equation.In the second chapter of the thesis,we consider the Dirichlet boundary value prob-lem for the prescribed mean curvature equation where ?>0,L>0 are parameters.By using the time map method,we obtain the exact number of positive solutions of the problem with three special forms of nonlinear term,that is,f(u)=e~u,f(u)=u(e~u-1),f(u)=e~u-1.The third chapter of the thesis consists of two parts.In the first part,by using bifurcation theory and extension theory,we study the existence of multiple positive solutions to the Neumann boundary value problem for the prescribed mean curvature equation where ? is parameter,and the validity of the obtained results are verified in examples.In the second part,we determine the interval of parameter ? in which the Neumann boundary value problem has exactly 2n+1 positive solutions by using time map method.The fourth chapter of the paper summarizes the research content and prospect.