Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent

In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type $$- mbox{div} big( a(x,u,nabla u)big)+ g(x,u,nabla u) + |u|^{p_{0}(x)-2}u = f-mbox{div} phi(u),quad mbox{ in } Omega,$$ where $-mbox{div}big(a(x,u,nabla u)big)$ is a Leray-Lions operator, $phi in C^{0}(I!!R,I!!R^{N})$. The function $g(x,u,nabla u)$ is a nonlinear lower order term with natural growth with respect to $|nabla u|$, satisfying the sign condition and the datum $f$ belongs to $L^1(Omega)$.