On Orlicz capacities and a nonexistence result for certain elliptic PDEs

  • Alberto FiorenzaEmail author
  • Flavia Giannetti


In this paper a new relation between a capacity of second order and an Orlicz capacity of first order is established. As a consequence, we construct a class of semilinear equations of second order, satisfying an assumption given in terms of first order capacities, which has no distributional solutions.


Orlicz capacity Measure data Cantor sets Elliptic equations 

Mathematics Subject Classification

31C15 46E35 35J60 


  1. 1.
    Adams, D.R., Hedberg, L.I.: Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314. Springer, Berlin (1996)Google Scholar
  2. 2.
    Aïssaoui N.: Note sur la capacitabilité dans les espaces d’Orlicz. Ann. Sci. Math. Québec 19(2), 107–113 (1995)zbMATHGoogle Scholar
  3. 3.
    Aïssaoui, N.: A Survey on Potential Theory on Orlicz Spaces, Recent Developments in Nonlinear Analysis, pp. 234–265. World Scientific Publication, Hackensack (2010)Google Scholar
  4. 4.
    Aïssaoui N., Benkirane A.: Capacités dans les espaces d’Orlicz. Ann. Sci. Math. Québec 18(1), 1–23 (1994)zbMATHGoogle Scholar
  5. 5.
    Aïssaoui N., Benkirane A.: Potentiel non linéaire dans les espaces d’Orlicz. Ann. Sci. Math. Québec 18(2), 105–118 (1994)zbMATHGoogle Scholar
  6. 6.
    Baras P., Pierre M.: Singularités éliminables pour des équations semi-linéaires. Ann. Inst. Fourier (Grenoble) 34(1), 185–206 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bidaut-Véron M.F.: Removable singularities and existence for a quasilinear equation with absorption or source term and measure data. Adv. Nonlinear Stud. 3, 25–63 (2003)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Fiorenza, A., Prignet, A.: Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data. ESAIM Control Optim. Calc. Var. 9, 317–341 (2003)Google Scholar
  9. 9.
    Gallouët T., Morel J.M.: Resolution of a semilinear equation in L 1. Proc. R. Soc. Edinb. Sect. A 96(3–4), 275–288 (1984)zbMATHCrossRefGoogle Scholar
  10. 10.
    Gallouët, T., Morel, J.M.: Corrigenda: “resolution of a semilinear equation in L 1”. Proc. R. Soc. Edinb. Sect. A 99(3–4), 399 (1985)Google Scholar
  11. 11.
    Krasnosel′skiĭ, M.A., Rutickiĭ, J.B.: Convex functions and Orlicz spaces, Translated from the first Russian edition by Boron, L.F., P. Noordhoff Ltd., Groningen (1961)Google Scholar
  12. 12.
    Malý J., Swanson D., Ziemer W.P.: Fine behavior of functions whose gradients are in an Orlicz space. Studia Math. 190(1), 33–71 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Marcus M.: Remarks on nonlinear equations with measures. Commun. Pure Appl. Anal. 12(4), 1745–1753 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Mizuta, Y., Ohno, T.: Orlicz capacity of balls. Complex analysis and potential theory, CRM Proc. Lecture Notes, vol. 55, pp. 225–233. Amer. Math. Soc., Providence (2012)Google Scholar
  15. 15.
    Nguyen Q.-H., Véron L.: Quasilinear and Hessian type equations with exponential reaction and measure data. Arch. Ration. Mech. Anal. 214, 235–267 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Orsina L., Prignet A.: Non-existence of solutions for some nonlinear elliptic equations involving measures. Proc. R. Soc. Edinb. Sect. A 130(1), 167–187 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Orsina L., Prignet A.: Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data. J. Funct. Anal. 189, 549–566 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Rao M.M., Ren Z.D.: Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker, New York (1991)Google Scholar
  19. 19.
    Véron L.: On the equation \({-\Delta u+{e}^{u}-1=0}\) with measures as boundary data. Math. Z. 273(1–2), 1–17 (2013)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Dipartimento di ArchitetturaUniversità di Napoli “Federico II”NapoliItaly
  2. 2.Istituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di Napoli Consiglio Nazionale delle RicercheNapoliItaly
  3. 3.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università di Napoli “Federico II”NapoliItaly

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