A Strongly Nonlinear Elliptic Problem with Generalized Growth in Musielak Spaces

In this article, we prove an existence theorem of renormalized solutions for nonlinear elliptic problem of the type -divA(x,u,∇u)-divΦ(x,u)+H(x,u,∇u)=finΩ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\mathop {\mathrm{div}}\>{\mathcal {A}}(x,u,\nabla u)-\mathop {\mathrm{div}}\varPhi (x,u)+{\mathcal {H}}(x,u,\nabla u)= f \quad \hbox {in }{\varOmega }, \end{aligned}$$\end{document}where the first lower-order term Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document} satisfies only a generalized natural growth condition without any supplementary assumptions. The approach does not require any particular type of growth condition on Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document}.