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This paper studies the eigenvalues of the p(x)-Laplacian Dirichlet problemwhere Ω is a bounded domain in R N and p(x) is a continuous function onΩ such that p(x) > 1. We show that Λ, the set of eigenvalues, is a nonempty infinite set such that sup Λ = +∞. We present some sufficient conditions for inf Λ = 0 and for inf Λ > 0, respectively. 

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We investigate Painlevé integrability of a generalized nonautonomous one-dimensional nonlinear Schrödinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painlevé analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painlevé integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.