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A weak solution to quasilinear elliptic problems with perturbed gradient

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Abstract

We consider weak solutions to the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}\,A\big (x,Du-\varTheta (u)\big )=f\quad &{}\text {in}\;\varOmega ,\\ u=0\quad &{}\text {on}\;\partial \varOmega , \end{array} \right. \end{aligned}$$

where \(\varTheta :{\mathbb {R}}^m\rightarrow {\mathbb {M}}^{m\times n}\) is a continuous function assumed to satisfy a Lipschitz condition. Based on the theory of Young measures, we prove the existence result when \(f\in W^{-1,p'}(\varOmega ;{\mathbb {R}}^m)\).

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Acknowledgements

We gratefully acknowledge the constructive comments of the referees concerning this paper.

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  1. Department of Mathematics, Faculty of Sciences Dhar El Mehraz, B.P. 1796, Atlas, Fez, Morocco

    Elhoussine Azroul & Farah Balaadich

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  1. Elhoussine Azroul
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  2. Farah Balaadich
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Correspondence to Farah Balaadich.

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Azroul, E., Balaadich, F. A weak solution to quasilinear elliptic problems with perturbed gradient. Rend. Circ. Mat. Palermo, II. Ser 70, 151–166 (2021). https://doi.org/10.1007/s12215-020-00488-4

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