Convergence Results of a Local Minimax Method for Finding Multiple Critical Points

In [14], a new local minimax method that characterizes a saddle point as a solution to a local minimax problem is established. Based on the local characterization, a numerical minimax algorithm is designed for finding multiple saddle points. Numerical computations of many examples in semilinear elliptic PDE have been successfully carried out to solve for multiple solutions. One of the important issues remains unsolved, i.e., the convergence of the numerical minimax method. In this paper, first, Step 5 in the algorithm is modified with the design of a new stepsize rule which is practically easier to implement and with which convergence results of the numerical minimax method are established for isolated and non-isolated critical points. The convergence results show that the algorithm indeed exceeds the scope of a minimax principle. In the last section, numerical multiple solutions to the Henon’s equation and a sublinear elliptic equation subject to zero Dirichlet boundary condition are presented to show their numerical convergence and profiles.