The Robin eigenvalue problem for the p(x)p(x)-Laplacian as p→∞p→∞

We study the asymptotic behavior, as p→∞p→∞, of the first eigenvalues and the corresponding eigenfunctions for the p(x)p(x)-Laplacian with Robin boundary conditions in an open, bounded domain Ω⊂RNΩ⊂RN with smooth boundary. We obtain uniform bounds for the sequence of first eigenvalues (suitably rescaled), and we prove that the positive first eigenfunctions converge uniformly in ΩΩ to a viscosity solution of a problem involving the∞∞-Laplacian subject to appropriate boundary conditions.