Entropy solutions for anisotropic unilateral elliptic problem with Neumann boundary conditions
Mohamed Badr Benboubker, Hayat Benkhalou, Hassane Hjiaj
Received March 13, 2024. Published online February 25, 2025.
Abstract: We consider the following strongly nonlinear Neumann elliptic problem: \begin{cases} \displaystyle-\sum^N_{i=1} D^i a_i(x,u,\nabla u) + H(x,u,\nabla u)+|u|^{p_0-2}u = f(x)+\sum^N_{i=1} D^i \phi_i(x,u) & in \Omega, \displaystyle\sum_{i=1}^N(a_i(x,u,\nabla u) - \phi_i(x,u))\cdot n_i= 0 & on \partial\Omega, \end{cases} where the Carathéodory functions $a_i(x,s,\xi),$ $H(x,s,\xi)$ and $\phi_i(x,s)$ verify some nonstandard conditions. By applying an approximation method, we prove the existence of entropy solutions for the unilateral problem with $L^1$-data, and we conclude some regularity results.
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Affiliations: Mohamed Badr Benboubker (corresponding author), High School of Technology, Sidi Mohamed Ben Abdellah University, Route Imouzzer, Fez, Morocco, e-mail: e-mail: simo.ben@hotmail.com, Hayat Benkhalou, Hassane Hjiaj, Department of Mathematics, Faculty of Sciences, Tetouan University Abdelmalek Essaadi, Quartier M'haneche II, Avenue Palestine, BP 2121, Tetouan 93000, Morocco, e-mail: benkhalouhayat@gmail.com, hjiajhassane@yahoo.fr
