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

On a PDE involving the ${\cal A}_{p(\cdot)}$-Laplace operator