  We study a class of nonlinear elliptic problems with Dirichlet conditions in
the framework of the Sobolev anisotropic spaces with variable exponent,
involving an anisotropic operator on an unbounded domain $\Omega\subset
\>I\!\!R^{N}\>(N \geq 2)\>$. We prove the existence of entropy solutions
avoiding sign condition and coercivity on the lowers order terms.
