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

Abstract

In this work, given p ∈ ( 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.