## 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

### Keywords:

Euler-Lagrange equation; nonlinear Rayleigh quotient
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