Multiplicity results for elliptic fractional equations with subcritical term

  • Giovanni Molica Bisci
  • Vicenţiu D. RădulescuEmail author


In the present paper, by using variational methods, we study the existence of multiple nontrivial weak solutions for parametric nonlocal equations, driven by the fractional Laplace operator \({(-\Delta)^{s}}\) , in which the nonlinear term has a sublinear growth at infinity. More precisely, a critical point result for differentiable functionals is exploited, in order to prove the existence of an open interval of positive eigenvalues for which the treated problem admits at least two nontrivial weak solutions in a suitable fractional Sobolev space.


Fractional Laplacian variational methods multiple solutions 

Mathematics Subject Classification

Primary: 49J35 35A15 35S15 Secondary: 47G20 45G05 


  1. 1.
    Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in \({\mathbb{R}^N}\). J. Differ. Equ. 255, 2340–2362 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barrios, B., Colorado, E., De Pablo, A., Sanchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133–6162 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brezis, H.: Analyse Fonctionnelle Théorie et Applications. Masson, Paris (1983)zbMATHGoogle Scholar
  4. 4.
    Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62(5), 597–638 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caffarelli, L., Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200(1), 59–88 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Capella, A.: Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains. Commun. Pure Appl. Anal. 10(6), 1645–1662 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dipierro, S., Pinamonti, A.: A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J. Differ. Equ. 255, 85–119 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fiscella, A., Servadei, R.: Valdinoci E A resonance problem for non-local elliptic operators. Z. Anal. Anwendungen (to appear)Google Scholar
  12. 12.
    Kristály, A., Radulescu, V., Varga, Cs.: Variational principles in mathematical physics, geometry, and economics. Qualitative analysis of nonlinear equations and unilateral problems. With a foreword by Jean Mawhin. In: Encyclopedia of Mathematics and its Applications, vol. 136. Cambridge University Press, Cambridge (2010)Google Scholar
  13. 13.
    Molica Bisci, G.: Fractional equations with bounded primitive. Appl. Math. Lett. 27, 53–58 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Molica Bisci, G.: Sequences of weak solutions for fractional equations. Math. Res. Lett. 21, 1–13 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Molica Bisci, G., Pansera, B.A.: Three weak solutions for nonlocal fractional equations. Adv. Nonlinear Stud. 14, 591–601 (2014)MathSciNetGoogle Scholar
  16. 16.
    Molica Bisci, G., Servadei R.: A bifurcation result for nonlocal fractional equations. Anal. Appl. (to appear). doi: 10.1142/S0219530514500067
  17. 17.
    Ricceri, B.: On a three critical points theorem. Archiv der Mathematik (Basel) 75, 220–226 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ricceri, B.: Existence of three solutions for a class of elliptic eigenvalue problems. Math. Comput. Model. 32, 1485–1494 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Servadei, R.: The Yamabe equation in a non-local setting. Adv. Nonlinear Anal. 2, 235–270 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Servadei, R.: Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity. Contemp. Math. 595, 317–340 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Servadei, R., Valdinoci, E.: Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoam. 29(3), 1091–1126 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33(5), 2105–2137 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Amer. Math. Soc. (to appear)Google Scholar
  25. 25.
    Tan, J.: The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc Var. Partial Differ. Equ. 36(1–2), 21–41 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Giovanni Molica Bisci
    • 1
  • Vicenţiu D. Rădulescu
    • 2
    Email author
  1. 1.Dipartimento PAUUniversità ‘Mediterranea’ di Reggio CalabriaReggio CalabriaItaly
  2. 2.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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