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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
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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.

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  • 1. S. Alama, M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. H. Poincaré Analyse Non Linéaire 13(1996) 95-115. MR 96m:35091
  • 2. S. Alama, G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calculus of Variations and Partial Differential Equations 1 (1993) 439-475. MR 97a:35057
  • 3. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18(1976) 620-709. MR 54:3519
  • 4. A. Ambrosetti, J. Garcia Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations, Jour. Funct. Anal. 137 (1996) 219-242. MR 97b:35059
  • 5. A. Ambrosetti, J. Garcia Azorero, I. Peral, Quasilinear equations with a multiple bifurcation, Differential Integral Equations, 10 (1997), 37-50. MR 97i:35036
  • 6. A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, Jour. Funct. Anal. 122(1994) 519-543. MR 95g:35059
  • 7. A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, Jour. Funct. Anal. 14(1973) 349-381. MR 51:6412
  • 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, Z.-Q. Wang, On the Morse indices of sign-changing solutions for nonlinear elliptic problems, Math. Zeit., 233 (2000), 655-677. MR 2001c:35079
  • 10. T. Bartsch, Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topological Meth. Nonlinear Anal., 7(1996) 115-131. MR 97m:35076
  • 11. H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 64(1991) 939-963. MR 92i:58032
  • 12. A. Castro, J. Cossio, J.M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mount. J. Math., 27 (1997), 1041-1053. MR 99f:35056
  • 13. A. Castro, J. Cossio, J.M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic BVPs, Electron. J. Diff. Equations (1998), No. 2, 18 pp. MR 98j:35060
  • 14. A. Castro, M. Finan, Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli, Topological Meth. Nonlinear Anal. 13(1999), 273-279. MR 2000j:35092
  • 15. K.C. Chang, A variant mountain pass lemma, Sci. Sinica Ser. A 26 (1983), no. 12, 1241-1255. MR 85h:58037
  • 16. K.C. Chang, Variational methods and sub- and supersolutions, Sci. Sinica Ser. A 26 (1983), no. 12, 1256-1265. MR 85h:58038
  • 17. K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser Boston 1993. MR 94e:58023
  • 18. K.C. Chang, Morse theory in nonlinear analysis. Nonlinear functional analysis and applications to differential equations (Trieste, 1997), 60-101, World Sci. Publishing, 1998. MR 2000k:58015
  • 19. G. Cerami, S. Solimini, M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69(1986), 289-306. MR 88b:35074
  • 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. D.C Clark, A variant of the Ljusternik-Schnirelman theory, Ind. Univ. Math. J. 22(1972) 65-74. MR 45:5836
  • 22. E.N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91(1983), 131-151. MR 84d:58020
  • 23. E.N. Dancer, Positivity of maps and applications, Topological Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 15(1995) 303-340, Birkhäuser, Boston. MR 95m:47119
  • 24. E.N. Dancer, Y. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), no. 3-4, 193-206. MR 97m:35061
  • 25. E.N. Dancer, Y. Du, Multiple solutions of some semilinear elliptic equations via the generalized Conley index, J. Math. Anal. Appl. 189 (1995), no. 3, 848-871. MR 96a:35059
  • 26. E.N. Dancer, Y. Du, The generalized Conley index and multiple solutions of semilinear elliptic problems, Abstract Appl. Anal., 1(1996) 103-135. MR 97i:35048
  • 27. E.N. Dancer, Y. Du, A note on multiple solutions of some semilinear elliptic problem, J. Math. Anal. Appl. 211(1997) 626-640. MR 98g:35075
  • 28. E.N. Dancer, S. Yan, On the profile of the changing sign mountain pass solutions for an elliptic problem, preprint.
  • 29. Z. Ding, D. Costa, G. Chen, A high-linking algorithm for sign-changing solutions of semilinear elliptic equations, Nonlinear Analysis, 38 (1999), 151-172. MR 2000d:65208
  • 30. E. Fadell, P.H. Rabinowitz, Generalized cohomological index thoeries for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Inv. Math., 45 (1978), 139-174. MR 57:17677
  • 31. J. Garcia Azorero, I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. AMS. 323 (1991), 877 - 895. MR 91g:35108
  • 32. H. P. Heinz, Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear systems, J. Diff. Eqns. 263 - 300 (1987). MR 88d:34020
  • 33. H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493 - 514. MR 84g:58030
  • 34. H. Hofer, A note on the topological degree at a critical point of mountain pass type, Proc. AMS 90(1984) 309-315. MR 85a:58015
  • 35. M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford, New York (1964). MR 28:2414
  • 36. S.J. Li, Some aspects of semilinear elliptic boundary value problem, Progress in Nonlinear Analysis, (1999) 234-256.
  • 37. Y. Li, Z.L. Liu, Multiple and sign-changing solutions of an elliptic eigenvalue problem with constraint, Science in China (Series A), 44(2001), No.1, 48-57. MR 2002b:35166
  • 38. S.J. Li, Z.-Q. Wang,. An abstract critical point theorem and applications, Acta Math. Sinica, 29(1986), 585-589. (Chinese) MR 88a:58036
  • 39. S.J. Li, Z.-Q. Wang, Mountain-pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. d'Analyse Math. 81(2000) 373-396. MR 2001h:35063
  • 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. J. M. Neuberger, A numerical method for finding sign-changing solutions of superlinear Dirichlet problems, Nonlinear World 4 (1997), 73-83. MR 98c:65200
  • 42. R.S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology, 5(1966), 1-16. MR 45:4184
  • 43. P. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partical differential equations, Ann. Scuola Norm. Sup. Pisa, Seri IV. 5 (1978) 215-223. MR 58:7695
  • 44. P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, R.I., (1986). MR 87j:58024
  • 45. G. Tarantello, Nodal solutions of semilinear elliptic equations with critical exponent, Differential Integral Equations 5(1992), 25-42. MR 92k:35109
  • 46. Z.-Q. Wang, A $Z_{p}$-index theory, Acta Math. Sinica (New Ser.), 6(1990), 18-23. MR 92a:58040
  • 47. Z.-Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Analyse Non Linéaire, 8(1991), 43 - 57. MR 92a:35064
  • 48. Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Diff. Equations Appl. 8(2001), 15-33. MR 2002c:35107
  • 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. M. Willem, Minimax Theorems, Birkhäuser (1996). MR 97h:58037

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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
Email: wang@math.usu.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03031-3
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