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} -\Delta_p u + (-\Delta)^r_p u & = & \frac{2\alpha}{\alpha+\beta}\lambda |u|^{\alpha-2}|v|^{\beta}u & \mbox{in} & \Omega \\ -\Delta_p v + (-\Delta)^s_p v& = & \frac{2\beta}{\alpha+\beta}\lambda |u|^{\alpha}|v|^{\beta-2}v & \mbox{in} & \Omega u& =& 0&\text{ on } & \mathbb{R}^N \setminus \Omega v& =& 0&\text{ on } & \mathbb{R}^N \setminus \Omega, \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.