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arXiv · 2402.10364

The Dirichlet problem for the $p(x)$-Laplacian with unbounded exponent $p(x)$

M. Khamsi·J. Lang·O. Mendez·A. Nekvinda
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Abstract

We prove the solvability of the Dirichlet problem for the variable exponent $p$-Laplacian with boundary data in $W^{1,p(x)}(\Omega)$ on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$. Our main focus will be on an a.e. finite variable exponent $p(\cdot)$ with $n < \inf\limits_{x\in \Omega}p(x)$ and $\sup\limits_{x\in \Omega}p(x) = \infty$ under the sole assumption that $p\in C(\Omega)$.

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M. Khamsi, J. Lang, O. Mendez, A. Nekvinda. 2024-02-15. The Dirichlet problem for the $p(x)$-Laplacian with unbounded exponent $p(x)$. https://arxiv.org/abs/2402.10364

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