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Entropy solutions for some nonlinear and noncoercive unilateral elliptic problems

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Abstract

This paper is concerned with the study of the existence results to the obstacle problem associated with the equation having degenerate coercivity, whose prototype is given by:

$$\begin{aligned} \left\{ \begin{array}{ll} -\hbox { div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^{p} = f(x,u) &{}\quad \hbox {in}\,\Omega ,\\ u = 0 &{}\quad \hbox {on}\,\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded open set of \(\mathbb {R}^N\) (\(N\ge 2\)), with \(1<p<N\), and \(f(\cdot ,s)\) satisfying some growth condition. We show the existence of entropy solutions for this non-coercive unilateral elliptic equation, and we will conclude some regularity results.

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Authors and Affiliations

  1. LAMA Laboratory, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, P.O. Box 1796, Atlas Fez, Morocco

    Youssef Akdim & Mounir Mekkour

  2. Laboratory LSI, Department of Mathematics, Physics and Informatics, Faculty Polydisciplinary of Taza, University Sidi Mohamed Ben Abdellah, P.O. Box 1223, Taza Gare, Morocco

    Mohammed Belayachi

  3. Department of Mathematics, Faculty of Sciences Tétouan, University Abdelmalek Essaadi, BP 2121, Tétouan, Morocco

    Hassane Hjiaj

Authors
  1. Youssef Akdim
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  2. Mohammed Belayachi
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  3. Hassane Hjiaj
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  4. Mounir Mekkour
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Correspondence to Mohammed Belayachi.

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Akdim, Y., Belayachi, M., Hjiaj, H. et al. Entropy solutions for some nonlinear and noncoercive unilateral elliptic problems. Rend. Circ. Mat. Palermo, II. Ser 69, 1373–1392 (2020). https://doi.org/10.1007/s12215-019-00477-2

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  • DOI: https://doi.org/10.1007/s12215-019-00477-2

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