Existence of an entropic solution of a nonlinear elliptic problem in an unbounded domain

We consider a second-order quasilinear elliptic equation with an integrable right-hand side. We formulate constraints on the structure of the equation in terms of a generalized N-function. We prove the existence of an entropic solution of the Dirichlet problem in nonreflexive Musielak-Orlicz-Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain.