On the Eigenvalues of the $p\&q-$ Fractional Laplacian

Abstract: We consider the eigenvalue problem for the fractional $p & q-$Laplacian \begin{equation} \left{\begin{aligned} (- \Delta)_p^{s}\, u + \mu(- \Delta)_q^{s}\, u+ |u|^{p-2}u+\mu|u|^{q-2}u=\lambda\ V(x)|u|^{p-2}u\quad & \text{in } \Omega\ u=0\quad& \text{in}\quad\R^N\backslash\Omega, \end{aligned}\right. \end{equation} where $\Omega$ is an open bounded, and possibly disconnected domain, $\lambda\in\R$, $1<q<p<\frac{N}{s}$, $\mu\>0$ with a weight function in $L^\infty(\Omega)$ that is allowed no change sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter $\mu\to 0^+.$ In addition, a stability property of eigenvalues as $s\to 1^-$ is established.