A minimax method for finding multiple critical points and its applications to semilinear PDEs (Q2780556)

The declared objective of the authors in this paper is to ``systematically develop effective numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way and not to establish new existence theorems.'' The first idea of the authors seems to have been a tentative to justify the very interesting mountain pass algorithm of \textit{Y. S. Choi} and \textit{P. J. McKenna} [Nonlinear Anal., Theory Methods Appl. 20, 417-437, (1993; Zbl 0779.35032)] and the high linking algorithm of \textit{Z. Ding}, \textit{D. Costa} and \textit{G. Chen} [Nonlinear Anal., Theory Methods Appl. 38A, 151-172, (1999; Zbl 0941.35023)]. They developed a local minimax method for finding multiple saddle points. This contrasts with the mountain pass theorem of \textit{A. Ambrosetti} and \textit{P. H. Rabinowitz} [J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)] characterizes a critical value as a (constrained) global maximization on compact sets at the first level and then a global minimization at the second level. NEWLINENEWLINENEWLINEThe authors point out that the solution obtained using the mountain pass algorithm may not be mathematically justified by the mountain pass theorem. This is due to global characterization of the critical level in the MPT (the minimum is taken over all paths joining 0 to the minimum \(e\)) while the algorithm will tend to the nearest solution to the maximum of the energy functional to the maximum of \(\Phi\) on \([0,e]\). We may also point out that until the moment, at the best knowledge of the reviewer, ``no error estimates'' has been given yet for the mountain pass algorithm. Moreover, in the case of boundary value problems that possess many solutions, the algorithm should not necessarily converge to the one characterized by the MPT. It should only converge to the nearest one to the maximum of the path used in the initialization (with a level greater than \(c = \inf_{\gamma\in\Gamma}\max_{\gamma([0,1])} \Phi\)). NEWLINENEWLINENEWLINEThe minimax method is used to solve some semilinear elliptic equations and some numerical examples for multiple solutions and their graphics are given.