On the logistic equation for the fractional p-Laplacian

TL;DR

This paper investigates a nonlinear nonlocal fractional p-Laplacian equation with logistic reaction, establishing existence, uniqueness, and bifurcation results across different diffusive regimes, and introduces a new comparison principle.

Contribution

It provides new existence and uniqueness results for the fractional p-Laplacian logistic equation and proves a bifurcation theorem using a novel comparison principle.

Findings

Existence and uniqueness of positive solutions in subdiffusive and equidiffusive cases.

Bifurcation results in the superdiffusive case.

Development of a new strong comparison principle.

Abstract

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.