Skip to main content
Log in

On local Morse theory for p-area functionals, p > 2

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

This survey article deals with some Morse theoretic aspects for functionals defined in Sobolev Banach spaces, associated with quasilinear elliptic equations or systems, involving the p-Laplace operator, p > 2.We discuss the notion of nondegeneracy in a Banach (not Hilbert) variational framework and we present some developments concerning the critical groups estimates and the interpretation of the multiplicity of a critical point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aftalion A., Pacella F.: Morse index and uniqueness for positive solutions of radial p-Laplace equations. Trans. Amer. Math. Soc. 356, 4255–4272 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alves C. O., Ding Y. H.: Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity. J. Math. Anal. Appl. 279, 508–521 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. García Azorero J. P., Peral Alonso I.: Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues. Comm. Partial Differential Equations 12, 1389–1430 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  4. García Azorero J., Peral Alonso I.: Multiplicity of solutions for elliptic problems with critical exponents or with a nonsymmetric term. Trans. Amer. Math. Soc. 323, 877–895 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. V. Benci, A new approach to the Morse-Conley theory and some applications. Ann. Mat. Pura Appl. (4) 158 (1991), 231–305.

    Google Scholar 

  6. Benci V., Cerami G.: Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc. Var. Partial Differential Equations 2, 29–48 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boccardo L., De Figueiredo D. G.: Some remarks on a system of quasilinear elliptic equations. NoDEA Nonlinear Differential Equations Appl. 9, 309–323 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Browder F.E.: Fixed point theory and nonlinear problems. Bull. Amer. Math. Soc. (N.S.) 9, 1–39 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carmona J., Cingolani S., Martínez-Aparicio P. J., Vannella G.: Regularity and Morse index of the solutions to critical quasilinear elliptic systems. Comm. Partial Differential Equations 38, 1675–1711 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chang K.C.: Morse theory on Banach space and its applications to partial differential equations. Chinese Ann. Math. Ser. B 4, 381–399 (1983)

    MATH  MathSciNet  Google Scholar 

  11. K. C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993.

  12. Cingolani S., Degiovanni M.: Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity. Comm. Partial Differential Equations 30, 1191–1203 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cingolani S., Degiovanni M.: On the Poincaré-Hopf theorem for functionals defined on Banach spaces. Adv. Nonlinear Stud. 9, 679–699 (2009)

    MATH  MathSciNet  Google Scholar 

  14. S. Cingolani, M. Degiovanni and B. Sciunzi, Critical groups estimates for p- Laplace equations via uniform Sobolev inequalities. Preprint submitted for publication.

  15. Cingolani S., Vannella G.: Critical groups computations on a class of Sobolev Banach spaces via Morse index. Ann. Inst. H. Poincaré Anal. Non Linéaire 20, 271–292 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Cingolani and G. Vannella, Morse index computations for a class of functionals defined in Banach space. In: Nonlinear Equations: Methods, Models and Applications (Bergamo, 2001), Progr. Nonlinear Differential Equations Appl. 54, Birkhäuser, Basel, 2003, 107–116.

  17. Cingolani S., Vannella G.: Morse index and critical groups for p-Laplace equations with critical exponents. Mediterr. J. Math. 3, 495–512 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Cingolani and G. Vannella, Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces. Ann. Mat. Pura Appl. (4) 186 (2007), 157–185.

    Google Scholar 

  19. Cingolani S., Vannella G.: Multiple positive solutions for a critical quasilinear equation via Morse theory. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 397–413 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Cingolani S., Vannella G.: On the multiplicity of positive solutions for p- Laplace equations via Morse theory. J. Differential Equations 247, 3011–3027 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cingolani S., Vannella G., Visetti D.: Morse index estimates for quasilinear equations on Riemannian manifolds. Adv. Differential equations 16, 1001–1020 (2011)

    MATH  MathSciNet  Google Scholar 

  22. S. Cingolani, G. Vannella and D. Visetti, Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds. Commun. Contemp. Math., to appear.

  23. Damascelli L., Sciunzi B.: Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Differential Equations 206, 483–515 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Degiovanni M.: On topological Morse theory. J. Fixed Point Theory Appl. 10, 197–218 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Degiovanni M., Lancelotti S.: Linking solutions for p-Laplace equations with nonlinearity at critical growth. J. Funct. Anal. 256, 3643–3659 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. de Morais Filho D. C., Souto M. A.: Systems of p-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees. Comm. Partial Differential Equations 24, 1537–1553 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. J. L. Diaz, Nonlinear Partial Differential Equations and Free Boundaries. I. Elliptic Equations. Res. Notes Math. 106, Pitman (Advanced Publishing Program), Boston, MA, 1985.

  28. Diaz J. I., de Thelin F.: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25, 1085–1111 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  29. DiBenedetto E.: C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ding L., Xiao S.-W.: Multiple positive solutions for a critical quasilinear elliptic system. Nonlinear Anal. 72, 2592–2607 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Glowinski R., Rappaz J.: Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM Math. Model. Numer. Anal. 37, 175–186 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Guedda M., Véron L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13, 879–902 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  33. Gromoll D., Meyer W.: On differentiable functions with isolated critical points. Topology 8, 361–369 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lancelotti S.: Morse index estimates for continuous functionals associated with quasilinear elliptic equations. Adv. Differential Equations 7, 99–128 (2002)

    MATH  MathSciNet  Google Scholar 

  35. Lieberman G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  36. Lindqvist P.: On nonlinear Rayleigh quotients. Potential Anal. 2, 199–218 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  37. Manasevich R., Mawhin J.: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differential Equations 145, 367–393 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  38. Marcellini P.: On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 391–409 (1986)

    MATH  MathSciNet  Google Scholar 

  39. A. Marino and G. Prodi, Metodi perturbativi nella teoria di Morse. Boll. Unione Mat. Ital. (4) 11 (1975), 1–32.

  40. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Appl. Math. Sci. 74, Springer-Verlag, New York, 1989.

  41. F. Mercuri and G. Palmieri, Problems in extending Morse theory to Banach spaces. Boll. Unione Mat. Ital. (4) 12 (1975), 397–401.

  42. Sciunzi B.: Some results on the qualitative properties of positive solutions of quasilinear elliptic equations. NoDEA Nonlinear Differential Equations Appl. 14, 315–334 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Smale S.: An infinite dimensional version of Sard's theorem. Amer. J. Math. 87, 861–866 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  44. E. H. Spanier, Algebraic Topology. McGraw-Hill, New York, 1966.

  45. Tromba A. J.: A general approach to Morse theory. J. Differential Geom. 12, 47–85 (1977)

    MATH  MathSciNet  Google Scholar 

  46. Uhlenbeck K.: Morse theory on Banach manifolds. J. Funct. Anal. 10, 430–445 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  47. Vélin J., de Thélin F.: Existence and nonexistence of nontrivial solutions for some nonlinear elliptic systems. Rev. Mat. Univ. Complut. Madrid 6, 153–194 (1993)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Silvia Cingolani.

Additional information

To Prof. Yvonne Choquet-Bruhat on the occasion of her 90th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cingolani, S. On local Morse theory for p-area functionals, p > 2. J. Fixed Point Theory Appl. 14, 355–373 (2013). https://doi.org/10.1007/s11784-014-0163-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-014-0163-6

Mathematics Subject Classification

Keywords

Navigation