  We study a Dirichlet boundary value problem associated to an anisotropic
differential operator on a smooth bounded of $\Bbb R^N$. Our main result
establishes the existence of at least two different non-negative solutions,
provided a certain parameter lies in a certain range. Our approach relies on
the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined
with adequate variational methods and a variant of Mountain Pass lemma.
