Regularity results for quasilinear elliptic problems driven by the fractional $\Phi$-Laplacian operator

TL;DR

This paper establishes $L^{p}$ estimates and boundedness of solutions for the fractional $ ext{ extPhi}$-Laplacian operator in bounded domains, covering a broad class of nonlinearities using Orlicz spaces and Moser's iteration.

Contribution

It provides new regularity results and $L^{ ext{infinity}}$ bounds for solutions of fractional $ ext{ extPhi}$-Laplacian problems in bounded domains, extending previous work to a wide class of nonlinearities.

Findings

Established $L^{p}$ estimates for fractional $ ext{ extPhi}$-Laplacian.

Proved solutions are bounded in $L^ ext{infinity}$ under certain conditions.

Applicable to a broad class of nonlinear operators and nonlinearities.

Abstract

It is established $L^{p}$ estimates for the fractional $\Phi$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the Moser's iteration, we prove that any weak solution for fractional $\Phi$-Laplacian operator defined in bounded domains belongs to $L^\infty(\Omega)$ under appropriate hypotheses on the $N$-function $\Phi$. Using the Orlicz space and taking into account the fractional setting for our problem the main results are stated for a huge class of nonlinear operators and nonlinearities.