Advertisement

On a class of nonhomogeneous fractional quasilinear equations in \({\mathbb{R}^n}\) with exponential growth

  • Manassés de SouzaEmail author
Article

Abstract

This paper examines a class of nonlocal equations involving the fractional p-Laplacian, where the nonlinear term is assumed to have exponential growth. More specifically, by using a suitable Trudinger–Moser inequality for fractional Sobolev spaces, we establish the existence of weak solutions for these equations.

Keywords

Fractional p-Laplacian Critical growth Trudinger–Moser inequality Fixed point result Discontinuous nonlinearity 

Mathematics Subject Classification

35J92 47H10 35B33 

References

  1. 1.
    Bartsch T., Wang Z.-Q.: Existence and multiplicity results for some superlinear elliptic problems on \({\mathbb{R}^N}\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011)Google Scholar
  3. 3.
    Caffarelli L.A.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symp. 7, 37–52 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carl S., Heikkilä S.: Elliptic problems with lack of compactness via a new fixed point theorem. J. Differ. Equ. 186, 122–140 (2002)CrossRefGoogle Scholar
  5. 5.
    Cheng, M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507 (2012)Google Scholar
  6. 6.
    de Oliveira, E.C., Vaz Jr, J.: Tunneling in fractional quantum mechanics. J. Phys. A 44, 185303 (2011)Google Scholar
  7. 7.
    de Souza M., do Ó J.M.: On a class of singular Trudinger–Moser type inequalities and its applications. Mathematische Nachrichten 284, 1754–1776 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    do Ó J.M., Medeiros E., Severo U.B.: A nonhomogeneous elliptic problem involving critical growth in dimension two. J. Math. Anal. Appl. 345, 286–304 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    do Ó J.M., Medeiros E., Severo U.B.: On a quasilinear nonhomogeneous elliptic equation with critical growth in \({\mathbb{R}^N}\). J. Differ. Equ. 246, 1363–1386 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Di Nezza E., Palatucci G., Valdinoci E.: Hitchhikers guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ding Y.H., Szulkin A.: Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. Partial Differ. Equ. 29, 397–419 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guo B., Huo Z.: Global well-posedness for the fractional nonlinear Schrödinger equation. Commun. Partial Differ. Equ. 36, 247–255 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Iannizzotto A., Squassina M.: 1/2-laplacian problems with exponential nonlinearity. J. Math. Anal. Appl. 414, 372–385 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kozono H., Sato T., Wadade H.: Upper bound of the best constant of a Trudinger–Moser inequality and its application to a Gagliardo–Nirenberg inequality. Indiana Univ. Math. J. 55, 1951–1974 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lam N., Lu G.: Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in \({\mathbb{R}^N}\). J. Funct. Anal. 262, 1132–1165 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Laskin N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)Google Scholar
  18. 18.
    Lindgren E., Lindqvist P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49, 795–826 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Moser J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)CrossRefGoogle Scholar
  20. 20.
    Nelson E.: Feynman integrals and the Schrödinger equation. J. Math. Phys. 5, 332–343 (1964)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ozawa T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127, 259–269 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rabinowitz P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Secchi, S.: (2013) Ground state solutions for nonlinear fractional Schrödinger equations in \({\mathbb{R}^N}\). J. Math. Phys. 54, 031501 (2013)Google Scholar
  24. 24.
    Sirakov B.: Existence and multiplicity of solutions of semi-linear elliptic equations in \({\mathbb{R}^N}\). Calc. Var. Partial Differ. Equ. 11, 119–142 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sirakov B.: Standing wave solutions of the nonlinear Schrödinger equation in \({\mathbb{R}^N}\). Ann. Mat. Pura Appl. (4) 181, 73–83 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Trudinger N.S.: On the imbedding into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Yang Y.: Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space. J. Funct. Anal. 262, 1679–1704 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang Z., Zhou H.S.: Ground state for nonlinear Schrödinger equation with sign-changing and vanishing potential. J. Math. Phys. 52, 113–704 (2011)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil

Personalised recommendations