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 W1,p(x)(Ω) on a bounded, smooth domain Ω ⊂ Rⁿ. Our main focus will be on an a.e. finite variable exponent p(·) with n < infx∈ Ωp(x) and x∈ Ωp(x) = ∞ under the sole assumption that p∈ C(Ω).