On the Eigenvalues of the $p\&q-$ Fractional Laplacian
Sabri Bahrouni, Hichem Hajaiej, Linjie Song · arXiv (Cornell University) · 2023
We consider the eigenvalue problem for the fractional $p \& q-$Laplacian \begin{equation} \left\{\begin{aligned} (- Δ)_p^{s}\, u + μ(- Δ)_q^{s}\, u+ |u|^{p-2}u+μ|u|^{q-2}u=λ V(x)|u|^{p-2}u\quad & \text{in } Ω\\ u=0\quad& \text{in}\quad\R^N\backslashΩ, \end{aligned}\right. \end{equation} where $Ω$ is an open bounded, and possibly disconnected domain, $λ\in\R$, $10$ with a weight function in $L^\infty(Ω)$ 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 $μ\to 0^+.$ In addition, a stability property of eigenvalues as $s\to 1^-$ is established.