## Ljusternik-Schnirelman theory in partially ordered Hilbert spaces

HTML articles powered by AMS MathViewer

- by Shujie Li and Zhi-Qiang Wang
- Trans. Amer. Math. Soc.
**354**(2002), 3207-3227 - DOI: https://doi.org/10.1090/S0002-9947-02-03031-3
- Published electronically: April 3, 2002
- PDF | Request permission

## 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.## References

- Stanley Alama and Manuel Del Pino,
*Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**13**(1996), no. 1, 95–115 (English, with English and French summaries). MR**1373473**, DOI 10.1016/S0294-1449(16)30098-1 - 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 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 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 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 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 10.1016/0022-1236(73)90051-7 - 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 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 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 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 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 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**, DOI 10.1007/978-1-4612-0385-8 - 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 10.1016/0022-1236(86)90094-7 - 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 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 10.1016/0022-247X(83)90098-7 - 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 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 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 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 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 10.1016/S0362-546X(98)00086-8 - 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 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 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 10.1016/0022-0396(87)90035-0 - Helmut Hofer,
*Variational and topological methods in partially ordered Hilbert spaces*, Math. Ann.**261**(1982), no. 4, 493–514. MR**682663**, DOI 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 10.1090/S0002-9939-1984-0727256-0 - M. A. Krasnosel’skii,
*Topological methods in the theory of nonlinear integral equations*, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak. 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 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 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 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**, DOI 10.1090/cbms/065 - 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 10.1007/BF02108859 - 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 - 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 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**, DOI 10.1007/978-1-4612-4146-1

## Bibliographic 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
- Received by editor(s): November 1, 2001
- Published electronically: April 3, 2002
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1897397