The ∞-Eigenvalue Problem

Abstract

. The Euler‐Lagrange equation of the nonlinear Rayleigh quotient $\left(\int_{\Omega}|\nabla u|^{p}\,dx\right) \bigg/ \left(\int_{\Omega}|u|^{p}\,dx\right)$ is $-\div\left( |\nabla u|^{p-2}\nabla u \right)= \Lambda_{p}^{p} |u |^{p-2}u,$ where $\Lambda_{p}^{p}$ is the minimum value of the quotient. The limit as $p\to\infty$ of these equations is found to be $\max \left\{ \Lambda_{\infty}-\frac{|\nabla u(x)|}{u(x)},\ \ \Delta_{\infty}u(x)\right\}=0,$ where the constant $\Lambda_{\infty}=\lim_{p\to\infty}\Lambda_{p}$ is the reciprocal of the maximum of the distance to the boundary of the domain Ω.