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Revised regularity results for quasilinear elliptic problems driven by the \(\Phi \)-Laplacian operator

  • E. D. Silva
  • M. L. CarvalhoEmail author
  • J. C. de Albuquerque
Article
  • 28 Downloads

Abstract

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known \(\Phi \)-Laplacian operator given by
$$\begin{aligned} \left\{ \ \begin{array}{ll} \displaystyle -\Delta _\Phi u= g(x,u), &{} \hbox {in}~\Omega ,\\ u=0, &{} \hbox {on}~\partial \Omega , \end{array} \right. \end{aligned}$$
where \(\Delta _{\Phi }u :=\hbox {div}(\phi (|\nabla u|)\nabla u)\) and \(\Omega \subset \mathbb {R}^{N}, N \ge 2,\) is a bounded domain with smooth boundary \(\partial \Omega \). Our work concerns on nonlinearities g which can be homogeneous or non-homogeneous. For the homogeneous case we consider an existence result together with a regularity result proving that any weak solution remains bounded. Furthermore, for the non-homogeneous case, the nonlinear term g can be subcritical or critical proving also that any weak solution is bounded. The proofs are based on Moser’s iteration in Orlicz and Orlicz-Sobolev spaces.

Mathematics Subject Classification

35B65 35B09 35D30 

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References

  1. 1.
    Adams, R.A., Fournier, J.F.: Sobolev Spaces. Academic Press, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Alves, C.O., Carvalho, M.L.M., Gonçalves, J.V.: On existence of solution of variational multivalued elliptic equations with critical growth via the Ekeland principle. Commun. Contemp. Math. 17(6), 1450038 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19(6), 581–597 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonanno, G., Molica Bisci, G., Rădulescu, V.: Existence of three solutions for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces. Topol. J. Nonlinear Anal. 74, 4785–4795 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonanno, G., Molica Bisci, G., Rădulescu, V.: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces. Topol. J. Math. Nonlinear Anal. 75, 4441–4456 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brezis, H., Kato, T.: Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. 58, 137–151 (1979)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cianchi, A.: Local boundedness of minimizers of anisotropic functionals. Ann. Inst. Henri Poincaré 17, 147–168 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Di Benedetto, E.: \(C^{1,\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. Theory Methods Appl. 7(8), 827–850 (1983)CrossRefGoogle Scholar
  9. 9.
    Fiscella, A., Pucci, P.: \((p, q)\) systems with critical terms in \(\mathbb{R}^N\). Special issue nonlinear PDEs and geometric function theory, in honor of Carlo Sbordone on his 70th birthday. Nonlinear Anal. 177(Part B), 454–479 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fucks, M., Gongbao, L.: \(L^\infty \)-bounds for elliptic equations on Orlicz–Sobolev spaces. Arch. Math. (Basel) 72(4), 293–297 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Annali di Matematica 186, 539–564 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fukagai, N., Ito, M., Narukawa, K.: Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on \(\mathbb{R}^{N}\). Funkcialaj Ekvacioj 49, 235–267 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gossez, J.P.: Nonlinear elliptic boundary value problems for equations with raplidy (or slowly) incressing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gossez, J.P.: Orlicz–Sobolev spaces and nonlinear elliptic boundary value problems. In: Nonlinear Analysis, Function Spaces and Applications (Proc. Spring School, Horni Bradlo, 1978), Teubner, Leipzig, pp. 59–94 (1979)Google Scholar
  15. 15.
    Ladyzenskaja, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)Google Scholar
  16. 16.
    Le, V.K.: A global bifurcation result for quasilinear elliptic equations in Orlicz–Sobolev spaces. Topol. Methods Nonlinear Anal. 15, 301–327 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva. In: Miniconference on Operators in Analysis, (Sydney, 1989), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 24, pp. 151–158. Austral. Nat. Univ., Canberra (1990)Google Scholar
  18. 18.
    Lou, H.: On singular sets of local solutions to \(p\)-Laplace equations. Chin. Ann. Math. 29B(5), 521–530 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mihăilescu, M., Rădulescu, V.: Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz–Sobolev space setting. Topol. J. Math. Anal. Appl. 330, 416–432 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Peral, I.: Multiplicity of solutions for the \(p\)-Laplacian. In: Second School of Nonlinear Functional Analysis and Applications to Differential Equations, International Center for Theoretical Physics Trieste (1997)Google Scholar
  21. 21.
    Pucci, P., Serrin, J.: The strong maximum principle revisited. J. Differ. Equ. 196, 1–66 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pucci, P., Servadei, R.: Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. Indiana Univ. Math. J. 57(7), 3329–3363 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rao, M.N., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1985)Google Scholar
  24. 24.
    Struwe, M.: Variational Methods Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3. Springer, Berlin (2000)Google Scholar
  25. 25.
    Tan, Z., Fang, F.: Orlicz–Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 402, 348–370 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • E. D. Silva
    • 1
  • M. L. Carvalho
    • 1
    Email author
  • J. C. de Albuquerque
    • 1
  1. 1.Department of MathematicsFederal University of GoiásGoiásBrazil

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