Research On The Calculation Method Of Multiple Solutions Of A Class Of Biharmonic Equations On The Unit Circle

The nonlinear fourth-order elliptic equation(Biharmonic equations)originated from the the-ory of elastic thin plates,which is widely used in Physics,Mechanics,Biology,Differential Ge-ometry and many other fields.In this dissertation,we mainly talk about the multiple solutions of the nonlinear fourth-order elliptic equation on a unit disk.The dissertation is divided into two sections:in the first section,we discuss the multiple solutions of Navier boundary value problem for a class of fourth-order elliptic equation on a unit disk.The form of this equation is(?)where ? is a unit disk,x is a two-dimensional vector,? ? R,q>1 and l>0 are given param-eters.First of all,we calculate the multiple non-trivial solutions of this equation(0.3)on a unit disk by finite difference method combining with the Liapunov-Schmidt reduction method and the symmetry-breaking bifurcation theory.Next,begining with the non-trivial solution branches of this nonlinear equation,we choose l in the problem(0.3)as the bifurcation parameter.Fol-lowed by numerically extending l,we further gain one symmetric positive solution branch of this nonlinear elliptic problem.During extending the parameter,we find the potential symmetry-breaking bifurcation points.Then we construct the expansion system which help to compute the symmetry-breaking bifurcation points accurately.We can also take advantage of the branch switching method in view of the Liapunov-Schmidt reduction to gain the multiple positive solu-tions which have various symmetrical character of the equation(0.3)on a unit disk.At last,we present the symmetry-breaking bifurcation positive solutions and the bifurcation diagrams with regard to the equation.In the second section,we discuss the multiple solutions of Dirichlet boundary value problem for the Biharmonic equations on a unit disk.The form of this equation is(?)where ? is a unit disk,x is a two-dimensional vector,l?0,q>1 and ??R and are given parameters.First of all,we calculate the multiple non-trivial solutions of this equation(0.4)on a unit disk by finite difference method combining with the Liapunov-Schmidt reduction method and the symmetry-breaking bifurcation theory.Next,begining with the non-trivial solution branches of this nonlinear equation,we choose l in the problem(0.4)as the bifurcation parameter.Followed by numerically extending l,we further gain one symmetric positive solution branch of this nonlinear elliptic problem.During extending the parameter,we find the potential symmetry-breaking bifurcation points.Then we construct the expansion system which help to compute the symmetry-breaking bifurcation points accurately.We can also take advantage of the branch switching method in view of the Liapunov-Schmidt reduction to gain the multiple positive solutions which have various symmetrical character of the equation(0.4)on a unit disk.At last,we present the symmetry-breaking bifurcation positive solutions and the bifurcation diagrams with regard to the equation.The numerical results confirm the truth of these approaches for proecessing the problems.The final section is for some conclusion discussions.