We discuss some aspects regarding the eigenvalue problem -Δp(x)u=λ|u|p(x)-2u if x∈Ω, u=0 if x∈∂Ω, where Ω⊂RN is a bounded domain, p:ˉΩ→(1,∞) is a continuous function and Δp(x)u:=∇⋅(|∇u|p(x)-2∇u) stands for the p(x)-Laplace operator. Let Λ be the set of eigenvalues of the above problem and λ*=inf. In particular, we will emphasize, on the one hand, situations when vanishes, and, on the other hand, we will advance some sufficient conditions when is positive. In the case when some extensions will be presented. In a related context some connections with a maximum principle will be pointed out.