An efficient and stable method for computing multiple saddle points with symmetries

SIAM Journal on Numerical Analysis

Volumn 43, Issue 2, 2005, Pages 891-907

  Wang, Zhi Qiang ^a   Jianxin, ^b  

^a UTAH STATE UNIVERSITY   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Invariance; Local minimax method; Morse index; Multiple saddle points; Semilinear PDE; Symmetry

Indexed keywords

LOCAL MINIMAX METHOD; MORSE INDEX; MULTIPLE SADDLE POINTS; SEMILINEAR PDE; SYMMETRY;

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL METHODS; ERROR ANALYSIS; INVARIANCE; ITERATIVE METHODS; PROBLEM SOLVING;

NUMERICAL METHODS;

EID: 33646477605     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903416626     Document Type: Article

References (25)

1. A. AMBROSETTI AND P. RABINOWITZ, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), pp. 349-381.
2. T. BARTSCH AND Z.-Q. WANG, On the existence of sign-changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), pp. 115-131.
3. H. BREZIS AND L. NIRENBERG, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), pp. 939-963.
4. J. BYEON AND Z.-Q. WANG, On the Hénon Equation: Asymptotic Profile of Ground States, preprint.
5. F. CATRINA AND Z.-Q. WANG, Nonlinear elliptic equations on expanding symmetric domains, J. Differential Equations, 156 (1999), pp. 153-181.
6. K.C. CHANG, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.
7. Y.S. CHOI AND P.J. MCKENNA, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Anal., 20 (1993), pp. 417-437.
8. D. COSTA, Z. DING, AND J. NEUBERGER, A numerical investigation of sign-changing solutions to superlinear elliptic equations on symmetric domains, J. Comput. Appl. Math., 131 (2001), pp. 299-319.
9. W.Y. DING AND W.M. NI, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91 (1986), pp. 238-308.
10. Z. DING, D. COSTA, AND G. CHEN, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Anal., 38 (1999), pp. 151-172.
11. S. Li AND Z.-Q. WANG, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), pp. 3207-3227.
12. Y. LI AND J. ZHOU, A minimax method for finding multiple critical points and its applications to semilinear PDEs, SIAM J. Sci. Comput. 23 (2001), pp. 840-865.
13. Y. LI AND J. ZHOU, Convergence results of a minimax method for finding multiple critical points, SIAM J. Sci. Comput., 24 (2002), pp. 865-885.
14. Y. LI AND J. ZHOU, Local characterizations of saddle points and their Morse indices, in Control Nonlinear Distributed Parameter Systems, G. Chen, I. Lasiecka, and J. Zhou, eds., Lecture Notes in Pure and Appl. Math. 218, Marcel Dekker, New York, 2001, pp. 233-251.
15. Z. LIU AND J. SUN, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), pp. 257-299.
16. J. MAWHIN AND M. WILLEM, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.
17. Z. NEHARI, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), pp. 101-123.
18. W.M. NI, Some Aspects of Semilinear Elliptic Equations, Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China, 1987.
19. W.M. NI, Recent Progress in Semilinear Elliptic Equations, RIMS Kokyuroku 679, Kyoto University, Kyoto, Japan, 1989, pp. 1-39.
20. P. RABINOWITZ, Minimax Method in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, AMS, Providence, RI, 1986.
21. M. SCHECHTER, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999.
22. M. STRUWE, Variational Methods, Springer-Verlag, New York, 1996.
23. Z.-Q. WANG, On a superlinear elliptic equation, Ann. Inst. H. Poincarè, 8 (1991), pp. 43-57.
24. M. WILLEM, Minimax Theorems, Birkhäuser, Boston, 1996.
25. J. ZHOU, Instability indices of saddle points by a local minimax method, Math. Comp., 74 (2005), pp. 1391-1411.