The Dirichlet problem for the $p(x)$-Laplacian with unbounded exponent $p(x)$

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