EXISTENCE OF A RENORMALIZED SOLUTION OF A QUASI-LINEAR ELLIPTIC EQUATION WITHOUT THE SIGN CONDITION ON THE LOWEST TERM

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The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are formulated in terms of the generalized 𝑁-function. Unlike the author’s previous works, there is no sign condition for the low-order term of the equation. In non-reflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain, the existence of a renormalized solution to the Dirichlet problem of this equation is proven.