Ljusternik-Schnirelman theory in partially ordered Hilbert spaces

Li, Shujie; Wang, Zhi-Qiang

doi:10.1090/S0002-9947-02-03031-3

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.

1. 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, https://doi.org/10.1016/S0294-1449(16)30098-1
2. 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, https://doi.org/10.1007/BF01206962
3. Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, https://doi.org/10.1137/1018114
4. 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, https://doi.org/10.1006/jfan.1996.0045
5. 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
6. 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, https://doi.org/10.1006/jfan.1994.1078
7. Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, https://doi.org/10.1016/0022-1236(73)90051-7
8. T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), 117-152. CMP 1 863 294
9. 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, https://doi.org/10.1007/s002090050492
10. 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, https://doi.org/10.12775/TMNA.1996.005
11. Haïm Brezis and Louis Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939–963. MR 1127041, https://doi.org/10.1002/cpa.3160440808
12. 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, https://doi.org/10.1216/rmjm/1181071858
13. 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},.
14. 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, https://doi.org/10.12775/TMNA.1999.014
15. Gong Qing Zhang, A variant mountain pass lemma, Sci. Sinica Ser. A 26 (1983), no. 12, 1241–1255. MR 745796
16. Gong Qing Zhang, Variational methods and sub- and supersolutions, Sci. Sinica Ser. A 26 (1983), no. 12, 1256–1265. MR 745797
17. 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
18. 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
19. 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, https://doi.org/10.1016/0022-1236(86)90094-7
20. 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. CMP 2001:01
21. David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972/73), 65–74. MR 296777, https://doi.org/10.1512/iumj.1972.22.22008
22. 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, https://doi.org/10.1016/0022-247X(83)90098-7
23. 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
24. 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, https://doi.org/10.1080/00036819508840321
25. 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, https://doi.org/10.1006/jmaa.1995.1054
26. 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, https://doi.org/10.1155/S108533759600005X
27. 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, https://doi.org/10.1006/jmaa.1997.5471
28. E.N. Dancer, S. Yan, On the profile of the changing sign mountain pass solutions for an elliptic problem, preprint.
29. 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, https://doi.org/10.1016/S0362-546X(98)00086-8
30. 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, https://doi.org/10.1007/BF01390270
31. 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, https://doi.org/10.1090/S0002-9947-1991-1083144-2
32. 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, https://doi.org/10.1016/0022-0396(87)90035-0
33. Helmut Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), no. 4, 493–514. MR 682663, https://doi.org/10.1007/BF01457453
34. 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, https://doi.org/10.1090/S0002-9939-1984-0727256-0
35. M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 0159197
36. S.J. Li, Some aspects of semilinear elliptic boundary value problem, Progress in Nonlinear Analysis, (1999) 234-256.
37. 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, https://doi.org/10.1007/BF02872282
38. 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
39. 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, https://doi.org/10.1007/BF02788997
40. 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.
41. 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
42. F.-H. Vasilescu, Local capacity of operators, Indiana Univ. Math. J. 21 (1971/72), 743–749. MR 295116, https://doi.org/10.1512/iumj.1972.21.21058
43. 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
44. 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
45. Gabriella Tarantello, Nodal solutions of semilinear elliptic equations with critical exponent, Differential Integral Equations 5 (1992), no. 1, 25–42. MR 1141725
46. 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, https://doi.org/10.1007/BF02108859
47. 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, https://doi.org/10.1016/S0294-1449(16)30276-1
48. 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, https://doi.org/10.1007/PL00001436
49. 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.
50. Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007