Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces

In this paper, we shall be concerned with the existence result of the following problem, 0.1 - div a ( x , u , ∇ u ) - div ( Φ ( x , u ) ) = f in Ω , u = 0 on ∂ Ω , with the second term f belongs to L 1 ( Ω ) . The growth and the coercivity conditions on the monotone vector field a are prescribed by a generalized N -function M . We assume any restriction on M , therefore we work with Musielak–Orlicz spaces which are not necessarily reflexive. The lower order term Φ is a Carathéodory function satisfying only a growth condition.