Entropy and renormalized solutions for a nonlinear elliptic problem in Musielak–Orlicz spaces

In this paper, we establish the equivalence of entropy and renormalized solutions of second-order elliptic equations with nonlinearities defined by the Musielak–Orlicz functions and the right-hand side from the space $L_1(\Omega).$ In nonreflexive Musielak—Orlicz—Sobolev spaces, we prove the existence and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with a Lipschitz boundary.