Existence of renomalized solution for nonlinear elliptic boundary value problem without $$\Delta _{2}$$ -condition

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In this paper we will prove in Musielak–Orlicz spaces, the existence of renomalized solution for nonlinear elliptic equations of Leray-Lions type, in the case where the Musielak–Orlicz function $$ \varphi $$ doesn’t satisfy the $$\Delta _{2}$$ condition while the right hand side f belongs to $$W^{-1} E_{\psi }(\Omega )$$ .