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

Abstract

In this article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \begin{align*} (-Δ)_p^αu - μ\frac{|u|^{p-2}u}{|x|^{pα}} = λV(x) |u|^{p-2}u \; \text{in}\; Ω, \quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ, \end{align*} where $n>p\alpha$, $p\ge 2$, $\alpha \in (0,1)$, $0\le \mu <C_{n,\alpha ,p}$ 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 $k\to \infty$.

30 pages