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Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

  • * Corresponding author: Thanh-Nhan Nguyen

    * Corresponding author: Thanh-Nhan Nguyen
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  • A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations $ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $ is established via Wolff type potentials. It is worthwhile to note that the model case of $ \mathbb{A} $ here is the non-degenerate $ p $-Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case $ 1<p \le \frac{3n-2}{2n-1} $, where the data $ \mu $ on right-hand side is assumed belonging to some classes that close to $ L^1 $. Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that $ \Omega $ is sufficiently flat in the Reifenberg sense.

    Mathematics Subject Classification: Primary: 35J62; 35J75; 35J92.

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