Some properties of the first eigenvalue of the $p(x)$-laplacian on riemannian manifolds

On the sub-supersolution principle for p(x)-Laplacian equations

“…For completeness, we first recall some facts on the variable exponent spaces L p(x) (Ω) and W 1,p(x) (Ω). For more details see [7,10,13,15,19]. Suppose that Ω is a bounded domain of R N with smooth boundary ∂Ω.…”