Preprint

Inserted: 19 jan 2025
Last Updated: 16 feb 2025

Year: 2025

Abstract:

The aim of this paper is to investigate the asymptotic behavior of the minimizers to the following problems related to the fractional $p-$Laplacian with nonhomogeneous term $h(u)$ in the presence of an obstacle $\psi$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$,

$ \min\bigg\{\frac{1}{2}\int_{\Omega \times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{\alpha p}} \,+\,\int_\Omega {h(u)}^p\,:\,u \in W^{s,p}(\Omega),\,\,u \geq \psi\,\,\,\mbox{on}\,\,\,\bar{\Omega},\,\, u=g\,\,\,\mbox{on}\,\,\partial\Omega\bigg\}. $

First, we show the convergence of the solutions to certain limit as $p \to \infty$ and identify the limit equation. More precisely, we show that the limit problem is closely related to the infinity fractional Laplacian. In the particular case when $h$ is increasing, we study the Holder regularity of any solution to the limit problem and we extend the existence result to the case when $h$ is singular.

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