Properties of eigenvalues and some regularities on fractional $p$-Laplacian with singular weights

TL;DR

This paper studies the fundamental properties and solution behaviors of fractional p-Laplacian eigenvalue problems with singular weights, establishing new results on eigenvalues, bounds, bifurcations, and solution existence.

Contribution

It introduces novel properties of eigenvalues and solutions for fractional p-Laplacian problems with singular weights, including bifurcation and existence results, even for classical Laplacian cases.

Findings

First eigenpair properties established

Existence of infinitely many solutions shown

Global bifurcation from the first eigenvalue demonstrated

Abstract

We provide fundamental properties of the first eigenpair for fractional $p$-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emenating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems involving singular weights. These are new results, even for (fractional) Laplacian.