A Minimax Method for Finding Multiple Critical Points and Its Applications to Semilinear PDEs

Most minimax theorems in critical point theory require to solve a two-level global optimization problem and therefore are not for algorithm implementation. The objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way. In this paper, inspired by the numerical works of Choi-McKenna and Ding-Costa-Chen, and the idea to define a solution submanifold, we establish some local minimax theorems, which require to solve only a two-level local optimization problem. Based on the local theory, a new local numerical minimax method for finding multiple saddle points is developed. The local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some non-convex, non star-shaped and multi-connected domains. Numerical solutions are illustrated by their graphics for visualization. In a subsequent paper [19], we establish some convergence results for the algorithm.