Instability analysis of saddle points by a local minimax method

Zhou, Jianxin

doi:10.1090/S0025-5718-04-01694-1

Instability analysis of saddle points by a local minimax method
HTML articles powered by AMS MathViewer

by Jianxin Zhou PDF

Math. Comp. 74 (2005), 1391-1411 Request permission

Abstract:

The objective of this work is to develop some tools for local instability analysis of multiple critical points, which can be computationally carried out. The Morse index can be used to measure local instability of a nondegenerate saddle point. However, it is very expensive to compute numerically and is ineffective for degenerate critical points. A local (weak) linking index can also be defined to measure local instability of a (degenerate) saddle point. But it is still too difficult to compute. In this paper, a local instability index, called a local minimax index, is defined by using a local minimax method. This new instability index is known beforehand and can help in finding a saddle point numerically. Relations between the local minimax index and other local instability indices are established. Those relations also provide ways to numerically compute the Morse, local linking indices. In particular, the local minimax index can be used to define a local instability index of a saddle point relative to a reference (trivial) critical point even in a Banach space while others failed to do so.

References

Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
A. Bahri and P.-L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), no. 8, 1027–1037. MR 968487, DOI 10.1002/cpa.3160410803
A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), no. 9, 1205–1215. MR 1177482, DOI 10.1002/cpa.3160450908
Thomas Bartsch and Zhi-Qiang Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 115–131. MR 1422008, DOI 10.12775/TMNA.1996.005
T. Bartsch, K.-C. Chang, and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z. 233 (2000), no. 4, 655–677. MR 1759266, DOI 10.1007/s002090050492
Haïm Brezis and Louis Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939–963. MR 1127041, DOI 10.1002/cpa.3160440808
Kung-ching Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1196690, DOI 10.1007/978-1-4612-0385-8
Y. S. Choi and P. J. McKenna, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Anal. 20 (1993), no. 4, 417–437. MR 1206432, DOI 10.1016/0362-546X(93)90147-K
Y. Chen and P. J. McKenna, Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations, J. Differential Equations 136 (1997), no. 2, 325–355. MR 1448828, DOI 10.1006/jdeq.1996.3155
Charles V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations 54 (1984), no. 3, 429–437. MR 760381, DOI 10.1016/0022-0396(84)90153-0
E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), no. 1, 120–156. MR 949628, DOI 10.1016/0022-0396(88)90021-6
Wei Yue Ding and Wei-Ming Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), no. 4, 283–308. MR 807816, DOI 10.1007/BF00282336
Zhonghai Ding, David Costa, and Goong Chen, A high-linking algorithm for sign-changing solutions of semilinear elliptic equations, Nonlinear Anal. 38 (1999), no. 2, Ser. A: Theory Methods, 151–172. MR 1697049, DOI 10.1016/S0362-546X(98)00086-8
J.J. Garcia-Ripoll, V.M. Perez-Garcia, E.A. Ostrovskaya and Y. S. Kivshar, Dipole-mode vector solitons, Phy. Rev. Lett., 85 (2000), 82-85.
Juan José García-Ripoll and Víctor M. Pérez-García, Optimizing Schrödinger functionals using Sobolev gradients: applications to quantum mechanics and nonlinear optics, SIAM J. Sci. Comput. 23 (2001), no. 4, 1316–1334. MR 1885603, DOI 10.1137/S1064827500377721
I. Kuzin and S. I. Pohozaev, Entire Solutions of Semilinear Elliptic Equations, Birkhauser, Boston, 1997.
A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal. 12 (1988), no. 8, 761–775. MR 954951, DOI 10.1016/0362-546X(88)90037-5
J. Q. Liu and S. J. Li, Some existence theorems on multiple critical points and their applications, Kexue Tongbao, 17 (1984).
Shu Jie Li and Michel Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), no. 1, 6–32. MR 1312028, DOI 10.1006/jmaa.1995.1002
Yongxin Li and Jianxin Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDEs, SIAM J. Sci. Comput. 23 (2001), no. 3, 840–865. MR 1860967, DOI 10.1137/S1064827599365641
Yongxin Li and Jianxin Zhou, Convergence results of a local minimax method for finding multiple critical points, SIAM J. Sci. Comput. 24 (2002), no. 3, 865–885. MR 1950515, DOI 10.1137/S1064827500379732
Yongxin Li and Jianxin Zhou, Local characterizations of saddle points and their Morse indices, Control of nonlinear distributed parameter systems (College Station, TX, 1999) Lecture Notes in Pure and Appl. Math., vol. 218, Dekker, New York, 2001, pp. 233–251. MR 1817184
Yan Yan Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations 83 (1990), no. 2, 348–367. MR 1033192, DOI 10.1016/0022-0396(90)90062-T
Jean Mawhin and Michel Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. MR 982267, DOI 10.1007/978-1-4757-2061-7
Z.H. Musslimani, M. Segev, D.N. Christodoulides and M. Soljacic, Composite Multihump vector solitons carrying topological charge, Phy. Rev. Lett., 84 (2000) 1164-1167.
Zeev Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101–123. MR 111898, DOI 10.1090/S0002-9947-1960-0111898-8
W.M. Ni, Some Aspects of Semilinear Elliptic Equations, Dept. of Math., National Tsing Hua Univ., Hsinchu, Taiwan, Rep. of China, 1987.
W.M. Ni, Recent progress in semilinear elliptic equations, in RIMS Kokyuroku 679, Kyoto University, Kyoto, Japan, 1989, 1-39.
Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785, DOI 10.1090/cbms/065
E. A. de B. e Silva, Multiple critical points for asymptotically quadratic functionals, Comm. Partial Differential Equations 21 (1996), no. 11-12, 1729–1770. MR 1421210, DOI 10.1080/03605309608821244
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Sergio Solimini, Morse index estimates in min-max theorems, Manuscripta Math. 63 (1989), no. 4, 421–453. MR 991264, DOI 10.1007/BF01171757
C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations 21 (1996), no. 9-10, 1431–1449. MR 1410836, DOI 10.1080/03605309608821233
Zhi Qiang Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 1, 43–57 (English, with French summary). MR 1094651, DOI 10.1016/S0294-1449(16)30276-1
Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1
J. Zhou, A min-orthogonal method for finding multiple saddle points., J. Math. Anal. Appl., 291 (2004), 66-81.