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

Abstract

Most minimax theorems in critical point theory require one 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, some local minimax theorems are established which require us 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 nonconvex, non star-shaped and multiconnected domains. Numerical solutions are illustrated by their graphics for visualization. In a subsequent paper [Y. Li and J. Zhou, Convergence results of a minimax method for finding critical points, in review], we establish some convergence results for the algorithm.