Remarks on eigenvalue problems involving the p(x)p(x)-Laplacian

This paper deals with the eigenvalue problem involving the p(x)p(x)-Laplacian of the form{−div(|∇u|p(x)−2∇u)=λ|u|q(x)−2uinΩ,u=0on∂Ω, where Ω   is a bounded domain in RNRN, p∈C0(Ω¯), infx∈Ωp(x)>1infx∈Ωp(x)>1, q∈L∞(Ω)q∈L∞(Ω), 1⩽q(x)⩽q(x)+ε0t>0, the problem has at least one sequence of solutions {(un,t,λn,t)}{(un,t,λn,t)} such that ∫Ω1p(x)|∇un,t|p(x)=t and λn,t→∞λn,t→∞ as n→∞n→∞. The principal eigenvalues for the problem in several important cases are discussed especially. The similarities and the differences in the eigenvalue problem between the variable exponent case and the constant exponent case are exposed. Some known results on the eigenvalue problem are extended.