  This paper establishes existence of solutions for a partial differential
equation in which a differential operator involving variable exponent growth
conditions is present. This operator represents a generalization of the
$p(\cdot)$-Laplace operator, i.e. $\Delta_{p(\cdot)}u={\rm div}(|\nabla
u|^{p(\cdot)-2}\nabla u)$, where $p(\cdot)$ is a continuous function. The proof
of the main result is based on Schauder's fixed point theorem combined with
adequate variational arguments. The function space setting used here makes
appeal to the variable exponent Lebesgue and Sobolev spaces.
