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Nonlinear Elliptic Systems with Coupled Gradient Terms

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Abstract

In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type:

$$ \left \{ \textstyle\begin{array}{r@{\quad }c@{\quad }l@{\quad }l@{\quad }l} -\Delta u & = & |\nabla v|^{q}+\lambda f & \text{in } &\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\mu g &\text{in } &\Omega , \\ u=v&=& 0 & \text{on } &\partial \Omega , \\ u,v& \geq & 0 & \text{in } &\Omega , \end{array}\displaystyle \right . $$

where \(\Omega \) is a bounded domain of \(\mathbb{R}^{N}\) and \(p, q\ge 1\). \(f,g\) are nonnegative measurable functions with additional hypotheses and \(\lambda , \mu \ge 0\).

This extends previous similar results obtained in the case where the right-hand sides are potential and gradient terms, see (Abdellaoui et al. in Appl. Anal. 98(7):1289–1306, [2019], Attar and Bentifour in Electron. J. Differ. Equ. 2017:1, [2017]).

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Acknowledgements

• The authors would like to thank

– the referee for the very careful reading of manuscript ;

– Prof. Boumediene Abdellaoui for his helpful suggestions and fruitful discussions during the preparation of this work.

• Part of this work was realized while the first author was visiting the Institut Elie Cartan, Université de Lorraine. He would like to thank the Institute for its warm hospitality.

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Authors and Affiliations

  1. Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen, 13000, Algeria

    Ahmed Attar & Rachid Bentifour

  2. Institut Elie Cartan, Université Lorraine, B.P. 239, 54506, Vandœuvre lés Nancy, France

    El-Haj Laamri

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  1. Ahmed Attar
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  2. Rachid Bentifour
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  3. El-Haj Laamri
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Correspondence to El-Haj Laamri.

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The first and the second authors are partially supported by DGRSDT, Algeria.

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Attar, A., Bentifour, R. & Laamri, EH. Nonlinear Elliptic Systems with Coupled Gradient Terms. Acta Appl Math 170, 163–183 (2020). https://doi.org/10.1007/s10440-020-00329-7

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