manuscripta mathematica

, Volume 155, Issue 3–4, pp 369–388 | Cite as

Multiplicity results for fractional Laplace problems with critical growth

  • Alessio Fiscella
  • Giovanni Molica Bisci
  • Raffaella Servadei
Article
First Online:
Received:
Accepted:
  • 130 Downloads

Abstract

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator \((-\Delta )^s\) and involving a critical Sobolev term. In particular, we consider
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su=\gamma \left| u\right| ^{2^*-2}u+f(x,u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$
where \(\Omega \subset \mathbb {R}^n\) is an open bounded set with continuous boundary, \(n>2s\) with \(s\in (0,1)\), \(\gamma \) is a positive real parameter, \(2^*=2n/(n-2s)\) is the fractional critical Sobolev exponent and f is a Carathéodory function satisfying different subcritical conditions. For this problem we prove two different results of multiple solutions in the case when f is an odd function. When f has not any symmetry it is still possible to get a multiplicity result: we show that the problem under consideration admits at least two solutions of different sign.

Mathematics Subject Classification

Primary: 49J35 35A15 35S15 Secondary: 47G20 45G05 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barrios, B., Colorado, E., Servadei, R., Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 875–900 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bartolo, R., Molica Bisci, G.: Asymptotically linear fractional \(p\)-Laplacian equations. Annali Mat. Pura Appl. (4) 196(2), 427–442 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brézis, H.: Analyse fonctionelle. Théorie et applications. Masson, Paris (1983)MATHGoogle Scholar
  6. 6.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, Bologna (2016). xii+155 pp. ISBN: 978-3-319-28738-6; 978-3-319-28739-3Google Scholar
  7. 7.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 7, 981–1012 (1983)MATHGoogle Scholar
  8. 8.
    Dipierro, S., Medina, M., Valdinoci, E.: Fractional Elliptic Problems with Critical Growth in the Whole of \(R^n\), Lecture Notes (Scuola Normale Superiore) 15, Edizioni della Normale, Pisa (2017). vii+158 pp. ISBN: 978-88-7642-601-8Google Scholar
  9. 9.
    Fiscella, A., Molica Bisci, G., Servadei, R.: Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems. Bull. Sci. Math. 140, 14–35 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fiscella, A., Pucci, P.: On certain nonlocal Hardy–Sobolev critical elliptic Dirichlet problems. Adv. Differ. Equ. 21, 571–599 (2016)MathSciNetMATHGoogle Scholar
  11. 11.
    Fiscella, A., Servadei, R., Valdinoci, E.: Density properties for fractional Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 40, 235–253 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5, 373–386 (2014)MathSciNetMATHGoogle Scholar
  14. 14.
    Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. 49, 795–826 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin, Encyclopedia of Mathematics and Its Applications, vol. 162, Cambridge University Press, Cambridge (2016). ISBN: 9781107111943Google Scholar
  16. 16.
    Molica Bisci, G., Servadei, R.: A bifurcation result for nonlocal fractional equations. Anal. Appl. 13(4), 371–394 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \(R^N\) involving nonlocal operators. Rev. Mat. Iberoam. 32, 1–22 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Servadei, R.: The Yamabe equation in a non-local setting. Adv. Nonlinear Anal. 2, 235–270 (2013)MathSciNetMATHGoogle Scholar
  20. 20.
    Servadei, R., Valdinoci, E.: Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoam. 29(3), 1091–1126 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discret. Contin. Dyn. Syst. 33(5), 2105–2137 (2013)MathSciNetMATHGoogle Scholar
  23. 23.
    Servadei, R., Valdinoci, E.: A Brezis–Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 12(6), 2445–2464 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Servadei, R., Valdinoci, E.: Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58(1), 133–154 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367(1), 67–102 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Silva, E.A.B., Xavier, M.S.: Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(2), 341–358 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Alessio Fiscella
    • 1
  • Giovanni Molica Bisci
    • 2
  • Raffaella Servadei
    • 3
  1. 1.Departamento de MatemáticaUniversidade Estadual de Campinas, IMECCCampinasBrazil
  2. 2.Dipartimento PAUUniversità ‘Mediterranea’ di Reggio CalabriaReggio CalabriaItaly
  3. 3.Dipartimento di Scienze Pure e Applicate (DiSPeA)Università degli Studi di Urbino ‘Carlo Bo’UrbinoItaly

Personalised recommendations