Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sign-changing critical points from linking type theorems


Authors: M. Schechter and W. Zou
Journal: Trans. Amer. Math. Soc. 358 (2006), 5293-5318
MSC (2000): Primary 35J20, 35J25, 58E05
DOI: https://doi.org/10.1090/S0002-9947-06-03852-9
Published electronically: January 24, 2006
MathSciNet review: 2238917
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, the relationships between sign-changing critical point theorems and the linking type theorems of M. Schechter and the saddle point theorems of P. Rabinowitz are established. The abstract results are applied to the study of the existence of sign-changing solutions for the nonlinear Schrödinger equation $ -\Delta u +V(x)u = f(x, u), u \in H^1({\mathbf{R}}^N),$ where $ f(x, u)$ is a Carathéodory function. Problems of jumping or oscillating nonlinearities and of double resonance are considered.


References [Enhancements On Off] (What's this?)

  • 1. T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), 117-152. MR 1863294 (2002i:58011)
  • 2. T. Bartsch, K. C. Chang and Z.-Q. Wang, On the Morse indices of sign-changing solutions for nonlinear elliptic problems, Math. Z. 233 (2000), 655-677. MR 1759266 (2001c:35079)
  • 3. H. Berestycki and de Figueiredo, Double resonance in semilinear elliptic problems, Comm. PDE 6 (1981), 91-120. MR 0597753 (82f:35078)
  • 4. T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), 25-42. MR 2038142 (2005d:35057)
  • 5. T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well. Comm. Contemp. Math. 4 (2001), 549-569.MR 1869104 (2002k:35079)
  • 6. T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal. 22 (2003), 1-14. MR 2037264 (2005d:35056)
  • 7. T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $ {\mathbf{R}}^N$. Comm. Partial Differential Equations 20 (1995), 1725-1741.MR 1349229 (96f:35050)
  • 8. T. Bartsch and Z.-Q. Wang, On the existence of sign-changing solutions for semilinear Dirichlet problems, Topo. Math. Nonlinear Anal. 7 (1996), 115-131. MR 1422008 (97m:35076)
  • 9. T. Bartsch, and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 13 (1999), 191-198. MR 1742220 (2001m:35096)
  • 10. H. Brézis, On a characterization of flow invariant sets, Comm. Pure. Appl. Math. 23 (1970), 261-263. MR 0257511 (41:2161)
  • 11. N. P. Cac, On nonlinear solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalues, J. Diff. Equ. 80 (1989), 397-404. MR 1011156 (90f:35077)
  • 12. K. C. Chang, Infinite-dimensional Morse theory and multiple solution problems, Birkhäuser, Boston, 1993. MR 1196690 (94e:58023)
  • 13. A. Castro, J. Cossio, and J. M. Neuberger, A sign changing solutions for a superlinear Dirichlet problems, Rocky Mount. J. Math. 27 (1997), 1041-1053. MR 1627654 (99f:35056)
  • 14. A. Castro, J. Cossio and J. M. Neuberger, A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic BVPs, E. J. Diff. Equations 2 (1999), 18pp. MR 1491525 (98j:35060)
  • 15. G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate. Rend. Acad. Sci. Let. Ist. Lombardo 112 (1978) 332-336.
  • 16. A. Castro and M. Finan, Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli, Topo. Meth. Nonlinear Anal. 13 (1999), 273-279. MR 1742224 (2000j:35092)
  • 17. K. Deimling, Ordinary differential equations in Banach spaces. Lecture Notes in Mathematics, Vol. 596. Springer-Verlag, Berlin-New York, 1977. MR 0463601 (57:3546)
  • 18. E. N. Dancer and Y. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), 193-206. MR 1383886 (97m:35061)
  • 19. E. N. Dancer and S. Yan, On the profile of the changing sign mountain pass solutions for an elliptic problem, Trans. Amer. Math. Soc. 354 (2002), 3573-3600. MR 1911512 (2003d:35082)
  • 20. S. Fucík, Boundary value problems with jumping nonlinearities, Casopis Pest. Mat. 101 (1976), 69-87. MR 0447688 (56:5998)
  • 21. A. C. Lazer and P. J. Mckenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, I Comm. PDE 10 (1985), 107-150. MR 0777047 (86f:35025)
  • 22. Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J. Differential Equations 172 (2001), 257-299.MR 1829631 (2002i:58012)
  • 23. S. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Tran. Amer. Math. Soc. 354 (2002), 3207-3227. MR 1897397 (2003c:58009)
  • 24. S. Li and 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 1785289 (2001h:35063)
  • 25. P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conf. Board of Math. Sci. Reg. Conf. Ser. in Math., No. 65, Amer. Math. Soc., 1986. MR 0845785 (87j:58024)
  • 26. P. Rabinowitz, On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992), 270-291. MR 1162728 (93h:35194)
  • 27. M. Schechter, Linking methods in critical point theory, Birkhäuser, Boston, 1999. MR 1729208 (2001f:58032)
  • 28. M. Schechter, Resonance problems with respect to the Fucík spectrum, Trans. Amer. Math. Soc. 352 (2000), 4195-4205. MR 1766536 (2001e:58012)
  • 29. M. Schechter, Critical points when there is no saddle point geometry, Topol. Methods Nonlinear Anal. 6 (1995), 295-308. MR 1399542 (97f:58033)
  • 30. M. Schechter, The Fucík spectrum, Indiana Univ. Math. J. 43 (1994), 1139-1157. MR 1322614 (96c:35063)
  • 31. M. Schechter, Critical points over splitting subspaces. Nonlinearity 6 (1993), 417-427. MR 1223741 (94e:58022)
  • 32. M. Schechter, The saddle point alternative, Amer. J. Math. 117 (1995), 1603-1626. MR 1363080 (96j:58033)
  • 33. M. Schechter and K. Tintarev, Pairs of critical points produced by linking subsets with applications to semilinear elliptic problems, Bull. Soc. Math. Belg. 44 (1994), 249-261. MR 1314040 (95k:58033)
  • 34. M. Struwe, Variational Methods, Springer-Verlag, Second Edition, 1996. MR 1411681 (98f:49002)
  • 35. D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonl. Anal. TMA. 23 (1994), 1401-1412. MR 1306679 (95i:35088)
  • 36. M. F. Furtado, L. A. Maia and E. A. B. Silva, On a double resonant problem in $ {\mathbf{R}}^N$. Differential Integral Equations 15 (2002), 1335-1344.MR 1920690 (2003g:35064)
  • 37. M. F. Furtado, L. A. Maia and E. A. B. Silva, Solutions for a resonant elliptic system with coupling in $ {\mathbf{R}}^N$. Comm. Partial Differential Equations 27 (2002), 1515-1536.MR 1924476 (2003f:35076)
  • 38. J. Sun, The Schauder condition in the critical point theory, Chinese Sci. Bull. 31 (1986), 1157-1162.MR 0866081 (88a:58038)
  • 39. E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal. 16 (1991), 455-477.MR 1093380 (92d:35108)
  • 40. E. A. B. Silva, Subharmonic solutions for subquadratic Hamiltonian systems. J. Differential Equations 115 (1995), 120-145. MR 1308608 (95k:58035)
  • 41. M. Schechter, Z-Q, Wang and W. Zou, New Linking Theorem and Sign-Changing Solutions, Comm. Partial Differential Equations 29 (2004), 471-488. MR 2041604 (2005c:35109)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J20, 35J25, 58E05

Retrieve articles in all journals with MSC (2000): 35J20, 35J25, 58E05


Additional Information

M. Schechter
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875

W. Zou
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9947-06-03852-9
Keywords: Sign-changing critical points, linking, jumping nonlinearities, oscillations, Schr\"{o}dinger equation, double resonance
Received by editor(s): June 9, 2003
Received by editor(s) in revised form: August 14, 2004
Published electronically: January 24, 2006
Additional Notes: The first authhor was supported by an NSF grant
The second author thanks the members of the Mathematics Department of the University of California at Irvine for an appointment to their department for the years 2001–2004. He was partially supported by NSFC10001019
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society