Abstract

We consider the eigenvalue problem for the fractional $p\&q$-Laplacian

where $\mathrm{\Omega }$ is an open, bounded, and possibly disconnected domain, $\lambda \in ℝ$, $1<q<p<N∕s$, $\mu >0$ with a weight function in $L^{\infty }(\mathrm{\Omega })$ that is allowed no change of sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter $\mu$ goes to $0^{+}$. In addition, a stability property of eigenvalues as $s\to 1^{-}$ is established.

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