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

In the paper,we study the multiple solution of boundary value problem for a class of nonlin-ear fourth-order elliptic problems(Biharmonic equations)on a unit square.The paper is divided into two parts:in the first part,we consider the multiple solution of Navier boundary value prob-lem for the Biharmonic equation on a unit square.The form of the Biharmonic equation is#12 where ? is a unit square,x is a two-dimensional vector,xo is the center point of the unit square,q>1,??R and r? 0 are given parameters.Firstly,we compute the multiple non-trivial solutions of the equation(0.1)on a unit square by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory,combined with Legendre-Galerkin spectral methods.Then,starting from the non-trivial solution branches of the corresponding nonlinear problem,we take r in the equation(0.1)as a bifurcation parameter and further obtain the symmetric positive solution branch of the equation(0.1)numerically by the r continuation method.During continuation,we find the potential symmetry-breaking bifurcation points.Then we propose the extended systems which can detect the symmetry-breaking bifurcation points on the positive solution branches accurately.We can compute the multiple positive solutions with various symmetries of the equation(0.1)on a unit square by the branch switching method based on the Liapunov-Schmidt reduction.Finally,we can show the symmetry-breaking bifurcation positive solutions of the equation(0.1)on a unit square and construct the bifurcation diagrams.In the second part,we consider the multiple solution of Drichlet boundary value problem for the Biarmonic equation on a unit square.The form of the Biharmonic equation is#12 where ? is a unit square,x is a two-dimensional vector,xo is the center point of the unit square,q>1,??R and r? 0 are given parameters.Firstly,we compute the multiple non-trivial solutions of the equation(0.2)on a unit square by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory,combined with Legendre-Galerkin spectral methods.Then,starting from the non-trivial solution branches of the corresponding nonlinear problem,we take r in the equation(0.2)as bifurcation parameter and further obtain the symmetric positive solution branch of the equation(0.2)numerically by the r continuation method.During continuation,we find the potential symmetry-breaking bifurcation points.Then we propose the extended systems which can detect the symmetry-breaking bifurcation points on the positive solution branches accurately.We can compute the multiple positive solutions with various symmetries of the equation(0.2)on a unit square by the branch switching method based on the Liapunov-Schmidt reduction.Finally,we can show the symmetry-breaking bifurcation positive solutions of the equation(0.2)on a unit square and construct the bifurcation diagrams.The numerical results demonstrate the effectiveness of these approaches.The final section is for some concluding discussions.