Eigenvalues of the p(x)p(x)-Laplacian Neumann problems

We study the eigenvalues of the p(x)p(x)-Laplacian operator with zero Neumann boundary condition on a bounded domain, where p(x)p(x) is a continuous function defined on the domain with p(x)>1p(x)>1. We show that, similarly to the pp-Laplacian case, the smallest eigenvalue of the problem is 0 and it is simple, and the supremum of all the eigenvalues is infinity, however, unlike the pp-Laplacian case, for very general variable exponent p(x)p(x), the first eigenvalue is not isolated, that is, the infimum of all positive eigenvalues of the problem is 0. We also study some properties of the set of functions having p(x)p(x)-average value zero.