Accepted Paper

Inserted: 19 jan 2025
Last Updated: 22 jul 2025

Journal: Forum Mathematicum
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 $h_p(x,u)$ in the presence of an obstacle $\psi$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$,

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

In the case when $h_p(x,u)=\frac{{h(x,u)}^p}{p}$ and $h(x,u) \geq 0$, 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 $\partial_s h >0$, 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. In addition, we will study the limit of this problem when the nonhomogeneous term $h_p(x,u)$ is not necessarily positive. To be more precise, we will consider the following two cases: $h_p(x,u)=h(x) u\,$ and $h_p(x,u)=h(x) \frac{\,\,|u|^{\Lambda}}{\Lambda}$ with $\Lambda:=\Lambda(p)<p$.

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• The limit of a nonlocal p-Laplacian obstacle problem