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On 1-Laplacian elliptic problems involving a singular term and an \(L^{1}\)-data

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Abstract

In this paper, we look at the problem

$$\begin{aligned} \left\{ \begin{array}{ll} {} -\Delta _{p}u+\vert \nabla u\vert ^{p}=fh(u) &{} \hbox { in } \Omega , \\ u\ge 0 &{} \hbox { in }\Omega , \\ u=0&{} \hbox { on } \partial \Omega , \end{array} \right. \end{aligned}$$

with \(\Omega \) is a bounded open subset of \(\mathbb {R}^{N}\) with Lipschitz boundary, \(\Delta _{p}u\) is the p-laplacian operator for \(1\le p<N, f\in L^{1}(\Omega )\) is nonnegative and h is a continuous function that may be singular at \(s=0^{+}.\) We will demonstrate the existence of solutions in the case \(1\le p<N.\) Moreover, if \(p=1, f>0\) and h is decreasing, we will show the uniqueness of the solutions.

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Acknowledgements

The authors would like to express their sincere gratitude to the referee for his/her valuable observations and suggestions.

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  1. Laboratory LIPIM, National School of Applied Sciences Khouribga, Sultan Moulay Slimane University, Khouribga, 25000, Morocco

    Youssef El Hadfi & Mohamed El Hichami

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  1. Youssef El Hadfi
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El Hadfi, Y., El Hichami, M. On 1-Laplacian elliptic problems involving a singular term and an \(L^{1}\)-data. J Elliptic Parabol Equ 9, 501–533 (2023). https://doi.org/10.1007/s41808-023-00210-2

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