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A Type of Brézis–Oswald Problem to the \({\Phi}\)- Laplacian Operator with Very Singular Term

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Abstract

In this work we consider existence and uniqueness of solutions for a quasilinear elliptic problem, which may be singular at the origin. Furthermore, we consider a comparison principle for subsolutions and supersolutions just in \({W^{1, \Phi}_{loc} (\Omega)}\) to the problem

$$\left\{\begin{array}{ll}-\Delta_{\Phi}u=f(x,u)\, {\rm in}\, \Omega,\\u > 0\, {\rm in} \, \Omega, u = 0 \,{\rm on}\, \partial\Omega,\end{array}\right.$$

where f has \({\Phi}\)-sublinear growth. In our main results the function f(x, u) may be singular at u =  0 and the nonlinear term \({f(x, t)/t^{\ell-1}, t > 0}\) is strictly decreasing for a suitable \({\ell > 1}\). Under different kind of boundary conditions we prove an improvement for the classical Brézis-Oswald and Díaz-Sáa’s results in Orlicz- Sobolev framework for singular nonlinearities as well. Some results discussed here are news even for Laplacian or p-Laplacian operators.

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Authors and Affiliations

  1. Universidade Federal de Goiás, Instituto de Matemática e Estatística, Goiânia, GO, 74001–970, Brazil

    M.L. Carvalho, J.V. Goncalves & E.D. Silva

  2. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910–900, Brazil

    C.A.P. Santos

Authors
  1. M.L. Carvalho
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  2. J.V. Goncalves
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  3. E.D. Silva
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  4. C.A.P. Santos
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Corresponding author

Correspondence to E.D. Silva.

Additional information

The authors were partially supported by Fapeg/CNpq grant 03/2015-PPP and CAPES/Brazil Proc. \({N^{o} \, 2788/2015 - 02}\).

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Carvalho, M., Goncalves, J., Silva, E. et al. A Type of Brézis–Oswald Problem to the \({\Phi}\)- Laplacian Operator with Very Singular Term. Milan J. Math. 86, 53–80 (2018). https://doi.org/10.1007/s00032-018-0279-z

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  • DOI: https://doi.org/10.1007/s00032-018-0279-z

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