Existence of a Renormalized Solution of a Quasilinear Elliptic Equation without the Sign Condition on the Lower-Order Term

Abstract

The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are stated in terms of the generalized \(N \)-function. Unlike the author’s previous papers, there is no sign condition on the lower-order term of the equation. The existence of a renormalized solution of the Dirichlet problem for this equation is proved in nonreflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain.