On the p(x)-Laplacian Robin eigenvalue problem

Consider Robin eigenvalue problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows:-Δp(x)u=λ|u|p(x)-2uinΩ,|∇u|p(x)-2∂u∂γ+β(x)|u|p(x)-2u=0on∂Ω.We prove the existence of infinitely many eigenvalue sequences if p(x) ≢ constant and also present some sufficient conditions for which there is no principal eigenvalue and the set of all eigenvalues is not closed.