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A note on fractional \({\varvec{p}}\)-Laplacian problems with singular weights

  • Ky Ho
  • Kanishka Perera
  • Inbo Sim
  • Marco Squassina
Article

Abstract

We study a class of fractional p-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related applications.

Keywords

Fractional p-Laplacian critical groups existence multiplicity 

Mathematics Subject Classification

35P15 35P30 35R11 

References

  1. 1.
    Applebaum, D.: Lévy processes, from probability fo finance and quantum groups. Not. Amer. Math. Soc. 51, 1336–1347 (2004)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Brasco, L., Franzina, G.: Convexity properties of Dirichlet integrals and Picone-type inequalities. Kodai Math. J. 37, 769–799 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brasco, L., Parini, E.: The second eigenvalue of the fractional \(p\)-Laplacian. Adv. Calc. Var. 9, 323–355 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Caffarelli, L.A.: Nonlocal equations, drifts and games. Non. Partial Diff. Eq. Abel Symposia 7, 37–52 (2012)Google Scholar
  5. 5.
    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. Henri Poincaré Nonlinear Anal. 31, 23–53 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dyda, B.: A fractional order Hardy inequality. Illinois J. Math. 48, 575–588 (2004)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Hashimoto, S., Otani, M.: Elliptic equations with singularity on the boundary. Diff. Integral Equ. 12, 339–349 (1999)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Hashimoto, S., Otani, M.: Sublinear elliptic equations and eigenvalue problems with singular coefficients. Commun. Appl. Anal. 6, 535–547 (2002)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Ladyzhenskaya, O., Uraltseva, N.: Linear and quasilinear elliptic equations. Academic Press, Dublin (1968)Google Scholar
  10. 10.
    Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fan, X., Han, X.: Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathbb{R}^N\). Nonlinear Anal. 59, 173–188 (2004)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5, 315–328 (2014)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Iannizzotto, A., Liu, S., Perera, K., Squassina, M.: Existence results for fractional \(p\)-Laplacian problems via Morse theory. Adv. Calc. Var. 9, 101–125 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. Rev. Mat. Iberoam (to appear)Google Scholar
  15. 15.
    Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional \(p\)-eigenvalue problems. Asymptot. Anal. 88, 233–245 (2014)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Ihnatsyeva, L., Lehrbäck, J., Tuominen, H., Vähäkangas, A.: Fractional Hardy inequalities and visibility of the boundary. Studia Math. 224, 47–80 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. PDE 49, 795–826 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Mosconi, S., Perera, K., Squassina, M., Yang, Y.: The Brezis–Nirenberg problem for the fractional \(p\)-Laplacian. Calc. Var. PDE 55, 105 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Perera, K.: Nontrivial critical groups in \(p\)-Laplacian problems via the Yang index. Topol. Methods Nonlinear Anal. 21, 301–309 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Perera, K., Agarwal, R.P., O’Regan, D.: Morse theoretic aspects of \(p\)-Laplacian type operators, 161 Mathematical Surveys and Monographs. Amer. Math. Soc, Providence, RI (2010)Google Scholar
  21. 21.
    Perera, K., Sim, I.: \(p\)-Laplace equations with singular weights. Nonlinear Anal. 99, 167–176 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ros-Oton, X., Serra, J.: The Pohožaev identity for the fractional Laplacian. Arch. Rat. Mech. Anal. 213, 587–628 (2014)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kajikiya, R.: Superlinear elliptic equations with singular coefficients on the boundary. Nonlinear Anal. 73, 2117–2131 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Amer. Math. Soc. 367, 67–102 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Usami, H.: On a singular elliptic boundary value problem in a ball. Nonlinear Anal. 13, 1163–1170 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)CrossRefzbMATHGoogle Scholar
  29. 29.
    Zhao, J.F.: Structure Theory of Banach Spaces. Wuhan University Press, Wuhan (1991)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Ky Ho
    • 1
  • Kanishka Perera
    • 2
  • Inbo Sim
    • 3
  • Marco Squassina
    • 4
  1. 1.NTISUniversity of West BohemiaPlzeňCzech Republic
  2. 2.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA
  3. 3.Department of MathematicsUniversity of UlsanUlsanRepublic of Korea
  4. 4.Dipartimento di Matematica e FisicaUniversità Cattolica del Sacro CuoreBresciaItaly

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