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

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.

MSC:

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