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Manuscripta Mathematica

, Volume 142, Issue 1–2, pp 157–185 | Cite as

Nonlinear elliptic problems on Riemannian manifolds and applications to Emden–Fowler type equations

  • Gabriele Bonanno
  • Giovanni Molica Bisci
  • Vicenţiu D. RădulescuEmail author
Article

Abstract

The existence of one non-trivial solution for a nonlinear problem on compact d-dimensional (\({d \geq 3}\)) Riemannian manifolds without boundary, is established. More precisely, a recent critical point result for differentiable functionals is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non-trivial weak solution. Moreover, as a consequence of our approach, a multiplicity result is presented, requiring the validity of the Ambrosetti–Rabinowitz hypothesis. Successively, the Cerami compactness condition is studied in order to obtain a similar multiplicity theorem in superlinear cases. Finally, applications to Emden-Fowler type equations are presented.

Mathematics Subject Classification (2010)

35J60 58J05 

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References

  1. 1.
    Aubin T.: Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aubin, T.: Nonlinear Analysis on Manifolds. Monge–Ampère Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252. Springer, New York (1982)Google Scholar
  3. 3.
    Aubin T.: Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin (1998)CrossRefGoogle Scholar
  4. 4.
    Beckner W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. Math. 138, 213–242 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bidaut-Véron M.F., Véron L.: Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. Invent. Math. 106, 489–539 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bonanno G.: Some remarks on a three critical points theorem. Nonlinear Anal. TMA 54, 651–665 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bonanno G.: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. 75, 2992–3007 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bonanno G., Molica Bisci G.: Three weak solutions for Dirichlet problems. J. Math. Anal. Appl. 382, 1–8 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bonanno G., Molica Bisci G., Rădulescu V.: Multiple solutions of generalized Yamabe equations on Riemannian manifolds and applications to Emden-Fowler problems. Nonlinear Anal. Real World Appl. 12, 2656–2665 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brézis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York (2011)Google Scholar
  11. 11.
    Cerami G.: Un criterio di esistenza per i punti critici su varietà illimitate. Rend. Inst. Lombardo Sci. Lett. 112, 332–336 (1978)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cotsiolis A., Iliopoulos D.: ’Equations elliptiques non linéaires sur \({\mathcal {S}^{n}}\) . Le problème de Nirenberg. C.R. Acad. Sci. Paris, Sér. I Math. 313, 607–609 (1991)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cotsiolis A., Iliopoulos D.: ’Equations elliptiques non linéaires à croissance de Sobolev sur-critique. Bull. Sci. Math. 119, 419–431 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ghoussoub N.: Duality and Perturbation Methods in Critical Point Theory Cambridge Tracts in Math. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  15. 15.
    Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics. AMS, New York (1999)Google Scholar
  16. 16.
    Hebey, E.: Variational Methods and Elliptic Equations in Riemannian Geometry. Notes from Lectures Given at ICTP, http://www.ucergy.fr/rech/pages/hebey/NotesICTP.pdf (2003)
  17. 17.
    Jeanjean L.: On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on \({\mathbb {R}^N}\) . Proc. R. Soc. Edinb. 129, 787–809 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kazdan J.L., Warner F.W.: Scalar curvature and conformal deformation of Riemannian structure. J. Differ. Geom. 10, 113–134 (1975)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kristály A.: Asymptotically critical problems on higher-dimensional spheres. Discr. Contin. Dyn. Syst. 23, 919–935 (2009)zbMATHCrossRefGoogle Scholar
  20. 20.
    Kristály A.: Bifurcation effects in sublinear elliptic problems on compact Riemannian manifolds. J. Math. Anal. Appl. 385, 179–184 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kristály A., Marzantowicz W.: Multiplicity of symmetrically distinct sequences of solutions for a quasilinear problem in \({\mathbb {R}^N}\) . Nonlinear Differ. Eqs. Appl. 15, 209–226 (2008)zbMATHCrossRefGoogle Scholar
  22. 22.
    Kristály A., Papageorgiou N., Varga C.S.: Multiple solutions for a class of neumann elliptic problems on compact Riemannian manifolds with boundary. Can. Math. Bull. Vol. 53, 674–683 (2010)zbMATHCrossRefGoogle Scholar
  23. 23.
    Kristály A., Rădulescu V.: Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations. Stud. Math. 191, 237–246 (2009)zbMATHCrossRefGoogle Scholar
  24. 24.
    Kristály, A., Rădulescu, V., Varga, C.S.: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, vol. 136. Cambridge University Press, Cambridge (2010)Google Scholar
  25. 25.
    Lee J.M., Parker T.H.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Liu S.: On superlinear problems without Ambrosetti and Rabinowitz condition. Nonlinear Anal. 73, 788–795 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Miyagaki O., Souto M.: Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equ. 245, 3628–3638 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Motreanu, D., Rădulescu, V.: Variational and non-variational methods in nonlinear analysis and boundary value problems. In: Nonconvex Optimization and Applications. Kluwer, Dordrecht (2003)Google Scholar
  29. 29.
    Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conferences in Mathematics, vol. 65. AMS, Providence (1985)Google Scholar
  31. 31.
    Ricceri B.: A general variational principle and some of its applications. J. Comput. Appl. Math. 113, 401–410 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Schoen R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Trudinger N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa 22, 265–274 (1968)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Vázquez J.L., Véron L.: Solutions positives d’équations elliptiques semi-linéaires sur des variétés riemanniennes compactes. C. R. Acad. Sci. Paris, Sér. I Math. 312, 811–815 (1991)zbMATHGoogle Scholar
  35. 35.
    Yamabe H.: On a deformation of Riemannian structures on compact manifolds. Osaka J. Math. 12, 21–37 (1960)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gabriele Bonanno
    • 1
  • Giovanni Molica Bisci
    • 2
  • Vicenţiu D. Rădulescu
    • 3
    • 4
    Email author
  1. 1.Mathematics Section, Department of Science for Engineering and Architecture, Engineering FacultyUniversity of MessinaMessinaItaly
  2. 2.Dipartimento MECMATUniversity of Reggio CalabriaReggio CalabriaItaly
  3. 3.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  4. 4.Department of MathematicsUniversity of CraiovaCraiovaRomania

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