A note on the eigenvalues of fractional Hardy-Sobolev operator with indefinite weight

TL;DR

This paper investigates the eigenvalues of a nonlinear fractional Hardy operator with an indefinite weight, establishing existence, simplicity, uniqueness, and asymptotic behavior of eigenvalues in a domain that may include singularities.

Contribution

It introduces new results on the eigenvalues of a fractional Hardy operator with indefinite weights, including simplicity, uniqueness, and the existence of an unbounded eigenvalue sequence.

Findings

The least positive eigenvalue is simple and unique with a non-negative eigenfunction.

A sequence of eigenvalues tends to infinity as the index increases.

The study covers operators with weights that can change sign and have singular points.

Abstract

In this article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \begin{align*} (-\Delta)_p^{\alpha}u - \mu \frac{|u|^{p-2}u}{|x|^{p\alpha}} = \lambda V(x) |u|^{p-2}u \; \text{in}\; \Omega, \quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminus\Omega, \end{align*} where $n>p\alpha$, $p\geq2$, $\alpha\in(0,1)$, $0\leq \mu <C_{n,\alpha,p}$ and $\Omega$ is a domain in $\mathbb{R}^n$ with Lipschitz boundary containing $0$. In particular, $\Omega=\mathbb{R}^n$ is admitted. The weight function $V$ may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is unique associated to a non-negative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues $\lambda_k \to \infty$ as $k\to\infty$.