$p(x)$-Stability of the Dirichlet problem for Poisson's equation with variable exponents

It is shown that if the sequence (pⱼ(x)) increases uniformly to $p(x)$ in a bounded, smooth domain $Ω$, then the sequence (uᵢ) of solutions to the Dirichlet problem for the pᵢ(x)-Laplacian with fixed boundary datum $φ$ converges (in a sense to be made precise) to the solution uₚ of the Dirichlet problem for the $p(x)$-Laplacian with boundary datum $φ$. A similar result is proved for a decreasing sequence pⱼ p