A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\to \infty$

Abstract

In this work, given $p\in (1,\infty )$, we prove the existence and simplicity of the first eigenvalue ${\lambda }_p$ and its corresponding eigenvector $(u_p,v_p)$, for the following local/nonlocal PDE system \begin{equation}\label{Eq0} \left\{ \begin{array}{rclcl} -Δ_p u + (-Δ)^r_p u & = & \frac{2α}{α+β}λ|u|^{α-2}|v|^βu & \mbox{in} & Ω\\ -Δ_p v + (-Δ)^s_p v& = & \frac{2β}{α+β}λ|u|^α|v|^{β-2}v & \mbox{in} & Ω u& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω v& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation} where $\Omega$$\subset$ ${\mathbb{R}}^N$ is a bounded open domain, $0<r,s<1$ and $\alpha (p)+\beta (p)=p$. Moreover, we address the asymptotic limit as $p\to \infty$, proving the explicit geometric characterization of the corresponding first $\infty -$eigenvalue, namely ${\lambda }_{\infty }$, and the uniformly convergence of the pair $(u_p,v_p)$ to the $\infty -$eigenvector $(u_{\infty },v_{\infty })$. Finally, the triple $(u_{\infty },v_{\infty },{\lambda }_{\infty })$ verifies, in the viscosity sense, a limiting PDE system.

References in corpus (1)

• Recent progresses in the theory of nonlinear nonlocal problems