Nonautonomous fractional problems with exponential growth

  • João Marcos do Ó
  • Olímpio H. Miyagaki
  • Marco Squassina


We study a class of nonlinear nonautonomous nonlocal equations with subcritical and critical exponential nonlinearity. The involved potential can vanish at infinity.


Trudinger-Moser inequality Schrödinger equations Vanishing potentials 

Mathematics Subject Classification

35P15 35P30 35R11 


  1. 1.
    Adams D.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 385–398 (1988)zbMATHCrossRefGoogle Scholar
  2. 2.
    Adimurthi A.: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian. Ann. Sci. Norm. Super. Pisa Cl. Sci. 17, 393–413 (1990)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Adimurthi, A., Yadava, S.L.: Multiplicity results for semilinear elliptic equations in a bounded domain of \({\mathbb{R}^2}\) involving critical exponent. Ann. Sci. Norm. Super. Pisa Cl. Sci., pp. 481–504 (1990)Google Scholar
  4. 4.
    Alves C.O., Souto M.A.S.: Existence of solutions for a class of elliptic equations in \({\mathbb{R}^N}\) with vanishing potentials. J. Differ. Equ. 252, 5555–5568 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Brandle C., Colorado E., Pablo A., Sánchez U.: A concave-convex elliptic problem involving the fractional laplacian. Proc. R. Soc. Edinb. A 143, 39–71 (2013)CrossRefGoogle Scholar
  6. 6.
    Cabré X., Tan J.G.: Positive solutions of nonlinear problems involving the square root of the laplacian. Adv. Math. 224, 2052–2093 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Caffarelli L., Silvestre L.: An extension problems related to the fractional laplacian. Commun. PDE 32, 1245–1260 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Carleson L., Chang A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 110, 113–127 (1986)zbMATHMathSciNetGoogle Scholar
  9. 9.
    do Ó, J.M., Miyagaki, O.H., Squassina, M.: Critical and subcritical fractional problems with vanishing potentials. preprintGoogle Scholar
  10. 10.
    Figueiredo D.G., do Ó J.M., Ruf B.: Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discret. Contin. Dyn. Syst. 30, 455–476 (2011)zbMATHCrossRefGoogle Scholar
  11. 11.
    Figueiredo D.G., Miyagaki O.H., Ruf B.: Elliptic equations in \({\mathbb{R}^2}\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equs. 3, 139–153 (1995)zbMATHCrossRefGoogle Scholar
  12. 12.
    Figueiredo D.G., do Ó J.M., Ruf B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math. 55, 135–152 (2002)zbMATHCrossRefGoogle Scholar
  13. 13.
    Frank R., Lenzmann E.: Uniqueness of non-linear ground states for fractional laplacians in \({\mathbb{R}}\) . Acta Math. 210, 261–318 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Iannizzotto A., Squassina M.: 1/2-Laplacian problems with exponential nonlinearity. J. Math. Anal. Appl. 414, 372–385 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Moser J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Ozawa T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127, 259–269 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Trudinger N.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • João Marcos do Ó
    • 1
  • Olímpio H. Miyagaki
    • 2
  • Marco Squassina
    • 3
  1. 1.Department of MathematicsFederal University of ParaíbaJoão PessoaBrazil
  2. 2.Department of MathematicsFederal University of Juiz de ForaJuiz de ForaBrazil
  3. 3.Dipartimento di InformaticaUniversità degli Studi di VeronaVeronaItaly

Personalised recommendations