Principal curves to fractional $m$-Laplacian systems and related maximum and comparison principles

TL;DR

This paper investigates principal eigenvalues and maximum principles for nonlinear systems involving fractional m-Laplacian operators, providing explicit bounds and conditions based on domain size.

Contribution

It introduces explicit lower bounds for principal eigenvalues and characterizes domain size conditions for maximum principles in fractional m-Laplacian systems.

Findings

Explicit lower bounds for principal eigenvalues in terms of domain diameter.

Conditions on domain size ensuring maximum principles hold.

Analysis of weak and strong maximum principles for nonlinear fractional systems.

Abstract

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional $m$-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of $\Omega$ are also proved. As application, given $\lambda,\mu\geq 0$ we measure explicitly how small has to be $\text{diam}(\Omega)$ so that weak and strong maximum principles associated to this problem hold in $\Omega$.