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

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


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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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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