Newton's method based on bifurcation for solving multiple solutions of nonlinear elliptic equations

On the basis of bifurcation theory, we use Newton's method to compute and visualize the multiple solutions to a series of typical semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in R2. We present three algorithms on the basis of the bifurcation method to solving these multiple solutions. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. The domains include the square, disk, symmetric or nonsymmetric annuli and dumbbell. The nonlinear partial differential equations include the Lane-Emden equation, concave-convex nonlinearities, Henon equation, and generalized Lane-Emden system. Copyright (c) 2013 John Wiley & Sons, Ltd.