A system of local/nonlocal p -Laplacians: The eigenvalue problem and its asymptotic limit as p→∞

In this work, given [Formula: see text], we prove the existence and simplicity of the first eigenvalue [Formula: see text] and its corresponding eigenvector [Formula: see text], for the following local/nonlocal PDE system [Formula: see text] where [Formula: see text] is a bounded open domain, [Formula: see text] and [Formula: see text]. Moreover, we address the asymptotic limit as [Formula: see text], proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely [Formula: see text], and the uniformly convergence of the pair [Formula: see text] to the ∞-eigenvector [Formula: see text]. Finally, the triple [Formula: see text] verifies, in the viscosity sense, a limiting PDE system.