Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

In this paper, we prove the existence and uniqueness of weak solution for a nonlinear degenerate Navier problem involving the weighted biharmonic operator of the following form: Δ[ϕ(z)a(z,Δw)]-div[ϑ1(z)K(z,∇w)+ϑ2(z)L(z,w,∇w)]+ϑ2(z)L0(z,w,∇w)=h0-∑j=1nDjhj,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} \Delta \Big [\phi (z)a(z,\Delta w)\Big ]-\mathrm{{div}}\Big [ \vartheta _{1}(z)\mathcal {K}(z,\nabla w)+\vartheta _{2}(z)\mathcal {L}(z,w,\nabla w)\Big ] \\{} & {} \qquad +\vartheta _{2}(z)\mathcal {L}_{0}(z,w,\nabla w)=h_0-\sum \limits _{j=1}^{n} D_{j}h_{j} \;, \end{aligned}$$\end{document}where ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}, ϑ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta _1$$\end{document} and ϑ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta _2$$\end{document} are weight functions, a:D¯×Rn⟶Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a:\overline{\mathcal {D}}\times \mathbb {R}^n\longrightarrow \mathbb {R}^n$$\end{document}, K:D×Rn⟶Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}:\mathcal {D}\times \mathbb {R}^n\longrightarrow \mathbb {R}^n$$\end{document}, L:D×R×Rn⟶Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}:\mathcal {D}\times \mathbb {R}\times \mathbb {R}^n\longrightarrow \mathbb {R}^n$$\end{document}, and L0:D×R×Rn⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_0:\mathcal {D}\times \mathbb {R}\times \mathbb {R}^n\longrightarrow \mathbb {R}$$\end{document} are Carathéodory applications that verified some conditions, and h0∈L1(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_0\in L^1(\mathcal {D})~$$\end{document} and hj∈Lp′(D,ϑ11-p′)(j=1,…,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_j\in L^{p'}(\mathcal {D},\vartheta _{1}^{1-p'})(j=1,\ldots ,n)$$\end{document}.