Convergence Results of A Minimax Method for Finding Multiple Critical Points

In [12], new local minimax theorems which characterize a saddle point as a solution to a two-level local minimax problem are established. Based on the local characterization, a numerical minimax method is designed for finding multiple saddle points. Many numerical 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, we first modify Step 5 in the algorithm such that it becomes practically easier to implement. Then we establish some convergence results for the numerical minimax method for isolated or non-isolated critical points. The convergence results show that the algorithm indeed exceeds the scope of a minimax principle.