Eigenvalues of p(x)-Laplacian Dirichlet problem

This paper studies the eigenvalues of the p(x)-Laplacian Dirichlet problem −div(|∇u|p(x)−2∇u)=λ|u|p(x)−2uinΩ,u=0on∂Ω, where Ω is a bounded domain in RN and p(x) is a continuous function on Ω̄ such that p(x)>1. We show that Λ, the set of eigenvalues, is a nonempty infinite set such that supΛ=+∞. We present some sufficient conditions for infΛ=0 and for infΛ>0, respectively.