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Israel Journal of Mathematics

, Volume 219, Issue 1, pp 331–351 | Cite as

A sharp eigenvalue theorem for fractional elliptic equations

  • Giovanni Molica Bisci
  • Vicenţiu D. Rădulescu
Article
  • 112 Downloads

Abstract

By using variational methods, in this paper we study a nonlinear elliptic problem defined in a bounded domain Ω ⊂ ℝ N , with smooth boundary ∂Ω, involving fractional powers of the Laplacian operator together with a suitable nonlinear term f. More precisely, we prove a characterization theorem on the existence of one weak solution for the elliptic problem
$$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^{\alpha /2}}\mu = \lambda f(\mu )in\Omega ,} \\ {u > 0in\Omega ,} \\ {u = 0in\partial \Omega ,} \end{array}} \right.$$
, where α ∈ (0, 2), N > α, λ > 0 and (−Δ)α/2 denotes the nonlocal fractional Laplacian operator. Our result extends to the nonlocal setting recent theorems for ordinary and classical elliptic equations, as well as a characterization for elliptic problems on certain non-smooth domains. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.

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References

  1. [1]
    G. Anello, A characterization related to the Dirichlet problem for an elliptic equation, Funkcialaj Ekvacioj 59 (2016), 113–122.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, Vol.116, Cambridge University Press, Cambridge, 2009.CrossRefMATHGoogle Scholar
  3. [3]
    G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in RN, Journal of Differential Equations 255 (2013), 2340–2362.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 32 (2015), 875–900.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    B. Barrios, E. Colorado, A. de Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, Journal of Differential Equations 252 (2012), 6133–6162.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Vol. 121, Cambridge University Press, Cambridge, 1996.MATHGoogle Scholar
  7. [7]
    C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 143 (2013), 39–71.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    H. Brezis, Analyse Fonctionelle. Théorie et Applications, Collection Mathématiques Appliqu ées pour la Maîtrise, Masson, Paris, 1983.Google Scholar
  9. [9]
    H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis 10 (1986), 55–64.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hemiltonian estimates, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 31 (2014), 23–53.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Mathematics 224 (2010), 2052–2093.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations 32 (2007), 1245–1260.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics 62 (2009), 597–638.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Archive for Rational Mechanics and Analysis 200 (2011), 59–88.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Communications on Pure and Applied Analysis 10 (2011), 1645–1662.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Communications in Partial Differential Equations 36 (2011), 1353–1384.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.MATHGoogle Scholar
  18. [18]
    G. Di Blasio and B. Volzone, Comparison and regularity results for the fractional Laplacian via symmetrization methods, Journal of Differential Equations 253 (2012), 2593–2615.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques 136 (2012), 521–573.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Communications in Mathematical Physics 337 (2015), 1317–1368.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Analysis & PDE 8 (2015), 57–114.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    G. Molica Bisci, Fractional equations with bounded primitive, Applied Mathematics Letters 27 (2014), 53–58.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    G. Molica Bisci, Sequences of weak solutions for fractional equations, Mathematical Research Letters 21 (2014), 1–13.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Advanced Nonlinear Studies 14 (2014), 591–601.MathSciNetMATHGoogle Scholar
  25. [25]
    G. Molica Bisci and V. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calculus of Variations and Partial Differential Equations 54 (2015), 2985–3008.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    G. Molica Bisci and V. Rădulescu, A characterization for elliptic problems on fractal sets, Proceedings of the American Mathematical Society 143 (2015), 2959–2968.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    G. Molica Bisci, V. Rădulescu and R. Servadei, 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.CrossRefMATHGoogle Scholar
  28. [28]
    G. Molica Bisci and D. Repovš, Higher nonlocal problems with bounded potential, Journal of Mathematical Analysis and Applications 420 (2014), 167–176.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    G. Molica Bisci and D. Repovš, Existence and localization of solutions for nonlocal fractional equations, Asymptotic Analysis 90 (2014), 367–378.MathSciNetMATHGoogle Scholar
  30. [30]
    G. Molica Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Analysis and Applications 13 (2015), 371–394.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    R. Musina and A. Nazarov, On fractional Laplacians, Communications in Partial Differential Equations 39 (2014), 1780–1790.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    P. Pucci and S. Saldi, Multiple solutions for an eigenvalue problem involving non-local elliptic p-Laplacian operators, in Geometric Methods in PDE’s, Springer INdAM Series, Vol. 13, Springer, Berlin, 2015, pp. 159–176.Google Scholar
  33. [33]
    P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in RN involving nonlocal operators, Revista Matemática Iberoamericana 32 (2016), 1–22.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    B. Ricceri, A note on spherical maxima sharing the same Lagrange multiplier, Fixed Point Theory and Applications 25 (2014), 1–9.MathSciNetMATHGoogle Scholar
  35. [35]
    B. Ricceri, A characterization related to a two-point boundary value problem, Journal of Nonlinear and Convex Analysis 16 (2015), 79–82.MathSciNetMATHGoogle Scholar
  36. [36]
    R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemporary Mathematics 595 (2013), 317–340.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    R. Servadei and E. Valdinoci, Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators, Revista Matemática Iberoamericana 29 (2013), 1091–1126.MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Transactions of the American Mathematical Society 367 (2015), 67–102.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 144 (2014), 1–25.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    J. Tan, The Brezis–Nirenberg type problem involving the square root of the Laplacian, Calculus of Variations and Partial Differential Equations 36 (2011), 21–41.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    B. Zhang, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity 28 (2015), 2247–2264.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  • Giovanni Molica Bisci
    • 1
  • Vicenţiu D. Rădulescu
    • 2
    • 3
  1. 1.Dipartimento PAUUniversità ‘Mediterranea’ di Reggio CalabriaReggio CalabriaItaly
  2. 2.Department of Mathematics, Faculty of SciencesKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of MathematicsUniversity of CraiovaCraiovaRomania

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