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*} (-\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 $k\to\infty$.