A minimax method for finding multiple critical points and its applicationsto semilinear PDEs

Most minimax theorems in critical point theory require one to solve a two-l evel global optimization problem and therefore are not for algorithm implem entation. The objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points i n a stable way. In this paper, inspired by the numerical works of Choi-McKe nna and Ding-Costa-Chen, and the idea to de ne a solution submanifold, some local minimax theorems are established which require us to solve only a tw o-level local optimization problem. Based on the local theory, a new local numerical minimax method for finding multiple saddle points is developed. T he local theory is applied, and the numerical method is implemented success fully to solve a class of semilinear elliptic boundary value problems for m ultiple solutions on some nonconvex, non star-shaped and multiconnected dom ains. Numerical solutions are illustrated by their graphics for visualizati on. In a subsequent paper [Y. Li and J. Zhou, Convergence results of a mini max method for finding critical points, in review], we establish some conve rgence results for the algorithm.