Eigenvalues of the p(x)p(x)-Laplacian Steklov problem

Consider Steklov eigenvalue problem involving the p(x)p(x)-Laplacian on a bounded domain Ω  , the open subset of RNRN with N⩾2N⩾2, as follows{Δp(x)u=|u|p(x)−2uinΩ,|∇u|p(x)−2∂u∂γ=λ|u|p(x)−2uon∂Ω, where p(x)≢p(x)≢constant.We prove that the existence of infinitely many eigenvalue sequences. Unlike the p  -Laplacian case, for a variable exponent p(x)p(x) (≢ constant), there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions. Finally, we present some sufficient conditions for the infimum of all eigenvalues is zero and positive, respectively.