Equivalence of Entropy and Renormalized Solutions of Nonlinear Elliptic Equations with $$L_{1}$$-Data in the Space $$\mathbb{R}^{n}$$

The paper considers elliptic equations of the second order with nonlinearities determined by the Musielak–Orlicz functions and the summable right-hand side. In non-reflexive Musielak–Orlicz–Sobolev spaces, the equivalence of entropy and renormalized solutions in the space $$\mathbb{R}^{n}$$ is proved and their existence and uniqueness are established.