Ljusternik-Schnirelman theory in partially ordered Hilbert spaces
Authors:
Shujie Li and Zhi-Qiang Wang
Journal:
Trans. Amer. Math. Soc. 354 (2002), 3207-3227
MSC (2000):
Primary 35J20, 35J25, 58E05
DOI:
https://doi.org/10.1090/S0002-9947-02-03031-3
Published electronically:
April 3, 2002
MathSciNet review:
1897397
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We present several variants of Ljusternik-Schnirelman type theorems in partially ordered Hilbert spaces, which assert the locations of the critical points constructed by the minimax method in terms of the order structures. These results are applied to nonlinear Dirichlet boundary value problems to obtain the multiplicity of sign-changing solutions.
- Stanley Alama and Manuel Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 1, 95–115 (English, with English and French summaries). MR 1373473, DOI https://doi.org/10.1016/S0294-1449%2816%2930098-1
- Stanley Alama and Gabriella Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), no. 4, 439–475. MR 1383913, DOI https://doi.org/10.1007/BF01206962
- Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI https://doi.org/10.1137/1018114
- Antonio Ambrosetti, Jesus Garcia Azorero, and Ireneo Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), no. 1, 219–242. MR 1383017, DOI https://doi.org/10.1006/jfan.1996.0045
- Antonio Ambrosetti, Jesus Garcia Azorero, and Ireneo Peral, Quasilinear equations with a multiple bifurcation, Differential Integral Equations 10 (1997), no. 1, 37–50. MR 1424797
- Antonio Ambrosetti, Haïm Brezis, and Giovanna Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543. MR 1276168, DOI https://doi.org/10.1006/jfan.1994.1078
- 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 https://doi.org/10.1016/0022-1236%2873%2990051-7
- T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), 117–152.
- 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 https://doi.org/10.1007/s002090050492
- 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 https://doi.org/10.12775/TMNA.1996.005
- Haïm Brezis and Louis Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939–963. MR 1127041, DOI https://doi.org/10.1002/cpa.3160440808
- Alfonso Castro, Jorge Cossio, and John M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), no. 4, 1041–1053. MR 1627654, DOI https://doi.org/10.1216/rmjm/1181071858
- Alfonso Castro, Jorge Cossio, and John M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electron. J. Differential Equations (1998), No. 02, 18 pp.}, review=\MR{1491525},.
- Alfonso Castro and Marcel B. Finan, Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli, Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 273–279. MR 1742224, DOI https://doi.org/10.12775/TMNA.1999.014
- Gong Qing Zhang, A variant mountain pass lemma, Sci. Sinica Ser. A 26 (1983), no. 12, 1241–1255. MR 745796
- Gong Qing Zhang, Variational methods and sub- and supersolutions, Sci. Sinica Ser. A 26 (1983), no. 12, 1256–1265. MR 745797
- 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
- K.-C. Chang, Morse theory in nonlinear analysis, Nonlinear functional analysis and applications to differential equations (Trieste, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 60–101. MR 1703528
- G. Cerami, S. Solimini, and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), no. 3, 289–306. MR 867663, DOI https://doi.org/10.1016/0022-1236%2886%2990094-7
- G. Chen, W. Ni, J. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations, Internat. Jour. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1565-1612.
- David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972/73), 65–74. MR 296777, DOI https://doi.org/10.1512/iumj.1972.22.22008
- E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), no. 1, 131–151. MR 688538, DOI https://doi.org/10.1016/0022-247X%2883%2990098-7
- E. N. Dancer, Positivity of maps and applications, Topological nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 15, Birkhäuser Boston, Boston, MA, 1995, pp. 303–340. MR 1322326
- E. N. Dancer and Yihong Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), no. 3-4, 193–206. MR 1383886, DOI https://doi.org/10.1080/00036819508840321
- E. N. Dancer and Yi Hong Du, Multiple solutions of some semilinear elliptic equations via the generalized Conley index, J. Math. Anal. Appl. 189 (1995), no. 3, 848–871. MR 1312556, DOI https://doi.org/10.1006/jmaa.1995.1054
- E. N. Dancer and Yihong Du, The generalized Conley index and multiple solutions of semilinear elliptic problems, Abstr. Appl. Anal. 1 (1996), no. 1, 103–135. MR 1390562, DOI https://doi.org/10.1155/S108533759600005X
- E. N. Dancer and Yihong Du, A note on multiple solutions of some semilinear elliptic problems, J. Math. Anal. Appl. 211 (1997), no. 2, 626–640. MR 1458519, DOI https://doi.org/10.1006/jmaa.1997.5471
- E.N. Dancer, S. Yan, On the profile of the changing sign mountain pass solutions for an elliptic problem, preprint.
- 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 https://doi.org/10.1016/S0362-546X%2898%2900086-8
- Edward R. Fadell and Paul H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), no. 2, 139–174. MR 478189, DOI https://doi.org/10.1007/BF01390270
- J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), no. 2, 877–895. MR 1083144, DOI https://doi.org/10.1090/S0002-9947-1991-1083144-2
- Hans-Peter Heinz, Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems, J. Differential Equations 66 (1987), no. 2, 263–300. MR 871998, DOI https://doi.org/10.1016/0022-0396%2887%2990035-0
- Helmut Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), no. 4, 493–514. MR 682663, DOI https://doi.org/10.1007/BF01457453
- Helmut Hofer, A note on the topological degree at a critical point of mountainpass-type, Proc. Amer. Math. Soc. 90 (1984), no. 2, 309–315. MR 727256, DOI https://doi.org/10.1090/S0002-9939-1984-0727256-0
- M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, The Macmillan Co., New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak; A Pergamon Press Book. MR 0159197
- S.J. Li, Some aspects of semilinear elliptic boundary value problem, Progress in Nonlinear Analysis, (1999) 234-256.
- Yongqing Li and Zhaoli Liu, Multiple and sign changing solutions of an elliptic eigenvalue problem with constraint, Sci. China Ser. A 44 (2001), no. 1, 48–57. MR 1828789, DOI https://doi.org/10.1007/BF02872282
- Shu Jie Li and Zhi Qiang Wang, An abstract critical point theorem and applications, Acta Math. Sinica 29 (1986), no. 5, 585–589 (Chinese). MR 876331
- Shujie Li and Zhi-Qiang Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math. 81 (2000), 373–396. MR 1785289, DOI https://doi.org/10.1007/BF02788997
- 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), no. 3, 840–865.
- John M. Neuberger, A numerical method for finding sign-changing solutions of superlinear Dirichlet problems, Nonlinear World 4 (1997), no. 1, 73–83. MR 1452506
- F.-H. Vasilescu, Local capacity of operators, Indiana Univ. Math. J. 21 (1971/72), 743–749. MR 295116, DOI https://doi.org/10.1512/iumj.1972.21.21058
- Paul H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 1, 215–223. MR 488128
- 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
- Gabriella Tarantello, Nodal solutions of semilinear elliptic equations with critical exponent, Differential Integral Equations 5 (1992), no. 1, 25–42. MR 1141725
- Zhi Qiang Wang, A $Z_p$ index theory, Acta Math. Sinica (N.S.) 6 (1990), no. 1, 18–23. A Chinese summary appears in Acta Math. Sinica 34 (1991), no. 2, 286. MR 1058898, DOI https://doi.org/10.1007/BF02108859
- Zhi Qiang Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 1, 43–57 (English, with French summary). MR 1094651, DOI https://doi.org/10.1016/S0294-1449%2816%2930276-1
- Zhi-Qiang Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 1, 15–33. MR 1828946, DOI https://doi.org/10.1007/PL00001436
- Z.-Q. Wang, Sign-Changing Solutions for a Class of Nonlinear Elliptic Problems, Nonlinear Analysis(Eds. K.-C. Chang and Y. Long), Nankai Series in Pure and Applied Math. 6 (2000), 370-383.
- Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007
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Additional Information
Shujie Li
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, P.R. China
Email:
lisj@math03.math.ac.cn
Zhi-Qiang Wang
Affiliation:
Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322
MR Author ID:
239651
Email:
wang@math.usu.edu
Keywords:
Ljusternik-Schnirelman theory,
order structure,
minimax method,
sign-changing solutions
Received by editor(s):
November 1, 2001
Published electronically:
April 3, 2002
Article copyright:
© Copyright 2002
American Mathematical Society