Advertisement

Journal of Mathematical Sciences

, Volume 236, Issue 4, pp 446–460 | Cite as

Multiplicity of Positive Solutions to the Boundary-Value Problems for Fractional Laplacians

  • N. S. Ustinov
Article

For the problem (−Δ)su=uq−1 in the annulus ΩR = BR+1 \ BR ∈ ℝn, a so-called “multiplicity effect” is established: for each N ∈ ℕ there exists R0 such that for all RR0 this problem has at least N different positive solutions. (−Δ)s in this problem stands either for Navier-type or for Dirichlet-type fractional Laplacian. Similar results were proved earlier for the equations with the usual Laplace operator and with the p-Laplacian operator.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Byeon, “Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli,” J. Diff. Eqs., 136, No. 1, 136–165 (1997).MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Caffarelli and L. Silvestre, “An extension problem related to the fractional Laplacian,” Comm. PDE’s, 32, No. 7–9, 1245–1260 (2007).MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Capella, J. Dávila, L. Dupaigne, and Y. Sire, “Regularity of radial extremal solutions for some non-local semilinear equations,” Comm. PDE’s, 36, No. 8, 1353–1384 (2011).MathSciNetCrossRefGoogle Scholar
  4. 4.
    C. V. Coffman, “A non-linear boundary-value problem with many positive solutions,” J. Diff. Eqs., 54, No. 3, 429–437 (1984).CrossRefGoogle Scholar
  5. 5.
    A. Cotsiolis and N. K. Tavoularis, “Best constants for Sobolev inequalities for higher order fractional derivatives,” J. Math. Anal. Appl., 295, No. 1, 225–236 (2004).MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Gazzola, H.-C. Grunau, and G. Sweers, “Optimal Sobolev and Hardy–Rellich constants under Navier boundary conditions,” Ann. Mat. Pura Appl. (4), 189, No. 3, 475–486 (2010).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Y. Ge, “Sharp Sobolev inequalities in critical dimensions,” Michigan Math. J., 51, No. 1, 27–45 (2003).MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Iannizzotto, S. Mosconi, and M. Squassina, “H s versus C 0-weighted minimizers,” NoDEA Nonlinear Diff. Eqs Appl., 22, No. 3, 477–497 (2015).CrossRefGoogle Scholar
  9. 9.
    S. B. Kolonitskii, “Multiplicity of solutions of the Dirichlet problem for an equation with the p-Laplacian in a three-dimensional spherical layer,” Algebra Analiz, 22, No. 3, 206–221 (2010).MathSciNetGoogle Scholar
  10. 10.
    A. Kufner and S. Fuchik, Nonlinear Differential Equations [Russian translation], Nauka, Moscow (1988).Google Scholar
  11. 11.
    Y. Y. Li, “Existence of many positive solutions of semilinear elliptic equations on annulus,” J. Diff. Eqs., 83, No. 2, 348–367 (1990).MathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Musina and A. I. Nazarov, “On fractional Laplacians,” Comm. PDE’s, 39, No. 9, 1780–1790 (2014).MathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Musina and A. I. Nazarov, “On the Sobolev and Hardy constants for the fractional Navier Laplacian,” Nonlinear Analysis, 121, 123–129 (2015).MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. Musina and A. I. Nazarov, “On fractional Laplacians–3,” J. ESAIM, 22, No. 3, 832–841 (2016).MathSciNetCrossRefGoogle Scholar
  15. 15.
    R. Musina and A. I. Nazarov, “Variational inequalities for the spectral fractional Laplacian,” Comp. Math. Math. Phys., 57, No. 3, 373–386 (2017).MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. I. Nazarov, “On solutions of the Dirichlet problem for an equation involving the p-Laplacian in a spherical layer,” Trudy St. Peterburg. Mat. Obshch., 10, 33–62 (2004).Google Scholar
  17. 17.
    R. S. Palais, “The principle of symmetric criticality,” Comm. Math. Phys., 69, No. 1, 19–30 (1979).MathSciNetCrossRefGoogle Scholar
  18. 18.
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [Russian translation], Mir, Moscow (1980).zbMATHGoogle Scholar
  19. 19.
    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces [in Russian], Mir, Moscow (1974).Google Scholar
  20. 20.
    P. R. Stinga and J. L. Torrea, “Extension problem and Harnack’s inequality for some fractional operators,” Comm. PDE’s, 35, No. 11, 2092–2122 (2010).MathSciNetCrossRefGoogle Scholar
  21. 21.
    R. C. A. M. Van der Vorst, “Best constant for the embedding of the space H 2 ∩ H 1 0(Ω) into L 2N/(N−4),” Diff. Integral Eqs,” 6, No. 2, 259–276 (1993).Google Scholar
  22. 22.
    G. N. Watson, A Treatise on the Theory of Bessel Functions. I [Russian translation], Moscow, Izd. Inostr. Lit. (1949).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations