The \(\infty\)-eigenvalue problem. (English) Zbl 0947.35104

Juutinen, Petri; Lindqvist, Peter; Manfredi, Juan J.

Arch. Ration. Mech. Anal. 148, No. 2, 89-105 (1999).

The authors prove the remarkable result that as \(p\to\infty\) the Euler-Lagrange equation of the nonlinear Rayleigh quotient becomes \[ \max\Biggl\{\Lambda_\infty- {|\nabla u(x)|\over u(x)}, \Delta_\infty u(x)\Biggr\}= 0, \] where \(\Lambda_\infty= {1\over\max\{\text{dist}(x, \partial\Omega;x\in\Omega)\}}\) and \(\Delta_\infty u(x)= \sum^n_{i,j= 1} {\partial u\over\partial x_i} {\partial u\over\partial x_j} {\partial^2u\over\partial x_i\partial x_j}\) is the \(\infty\)-Laplacian.

Reviewer: B.D.Sleeman (Leeds)

MSC:

35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

Euler-Lagrange equation; nonlinear Rayleigh quotient