Bifurcation method for solving multiple positive solutions to Henon equation

Three algorithms based on the bifurcation method are applied to solving the D ₄ symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bifurcation parameter, the D ₄−Σ _d (D ₄ − Σ₁, D ₄ − Σ₂) symmetry-breaking bifurcation points on the branch of the D ₄ symmetric positive solutions are found via the extended systems. Finally, Σ _d (Σ₁, Σ₂) symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction. This work was supported by the National Natural Science Foundation of China (Grant No. 10671130), the Science Foundation of Shanghai Municipal Education Commission (Grant No. 05DZ07), Shanghai Leading Academic Discipline Project (Grant No. T0401), Leading Foundation of Shanghai Science and Technology Commission (Grant No. 06JC14092) and the Foundation of the Scientific Computing Key Laboratory of Shanghai Universities