  We prove the solvability of the Dirichlet problem for the variable exponent
$p$-Laplacian with boundary data in $W^{1,p(x)}(\Omega)$ on a bounded, smooth
domain $\Omega \subset {\mathbb R}^n$. Our main focus will be on an a.e. finite
variable exponent $p(\cdot)$ with $n < \inf\limits_{x\in \Omega}p(x)$ and
$\sup\limits_{x\in \Omega}p(x) = \infty$ under the sole assumption that $p\in
C(\Omega)$.
