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

In this work, given p ϵ (1, ∞), we prove the existence and simplicity of the first eigenvalue λ p and its corresponding eigenvector (u p, v p), for the following local/nonlocal PDE system (0.1)-δ p u + (-δ) p r u = 2 α α + β λ | u | α-2 | v | β u in ω-δ p v + (-δ) p s v = 2 β α + β λ | u | α | v | β-2 v in ω u = 0 on R N-ω v = 0 on R N-ω, where ω R N is a bounded open domain, 0 < r, s < 1 and α (p) + β (p) = p. Moreover, we address the asymptotic limit as p → ∞, proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely λ ∞, and the uniformly convergence of the pair (u p, v p) to the ∞-eigenvector (u ∞, v ∞). Finally, the triple (u ∞, v ∞, λ ∞) verifies, in the viscosity sense, a limiting PDE system.