# Sobolev versus H\"older minimizers for the degenerate fractional $p$-Laplacian

## TL;DR

This paper investigates the relationship between Sobolev and H"older minimizers for a degenerate fractional p-Laplacian problem, showing they coincide under certain conditions, which advances understanding of minimizer regularity.

## Contribution

It establishes the equivalence of Sobolev and H"older local minimizers for the degenerate fractional p-Laplacian, connecting different regularity frameworks.

## Key findings

- Sobolev and H"older minimizers coincide for the problem
- Provides conditions under which minimizers in different spaces are equivalent
- Advances understanding of regularity and minimizer behavior in fractional p-Laplacian problems

## Abstract

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-\Delta)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $\Omega$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(\Omega)$ and in the weighted H\"older space $C^0_s(\overline\Omega)$, respectively, do coincide.