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: \(\begin{aligned}{} & {} \Delta \Big [\phi (z)a(z,\Delta w)\Big ]-\mathrm{{div}}\Big [ \vartheta _{1}(z)\mathcal {K}(z,\nabla w)+\vartheta _{2}(z)\mathcal {L}(z,w,\nabla w)\Big ] \\{} & {} \qquad +\vartheta _{2}(z)\mathcal {L}_{0}(z,w,\nabla w)=h_0-\sum \limits _{j=1}^{n} D_{j}h_{j} \;, \end{aligned}\) $$\begin{array}{cc}\hfill & \mathrm{\Delta }[\varphi (z)a(z,\mathrm{\Delta }w)]-\mathrm{div}[{\vartheta }_1(z)\mathcal{K}(z,\mathrm{\nabla }w)+{\vartheta }_2(z)\mathcal{L}(z,w,\mathrm{\nabla }w)]\hfill \\ \hfill & +{\vartheta }_2(z){\mathcal{L}}_0(z,w,\mathrm{\nabla }w)=h_0-\sum _{j=1}^n{D}_j{h}_j,\hfill \end{array}$$ where \(\phi \) $\varphi$ , \(\vartheta _1\) ${\vartheta }_1$ and \(\vartheta _2\) ${\vartheta }_2$ are weight functions, \(a:\overline{\mathcal {D}}\times \mathbb {R}^n\longrightarrow \mathbb {R}^n\) $a:\overline{\mathcal{D}}\times {\mathbb{R}}^n⟶{\mathbb{R}}^n$ , \(\mathcal {K}:\mathcal {D}\times \mathbb {R}^n\longrightarrow \mathbb {R}^n\) $\mathcal{K}:\mathcal{D}\times {\mathbb{R}}^n⟶{\mathbb{R}}^n$ , \(\mathcal {L}:\mathcal {D}\times \mathbb {R}\times \mathbb {R}^n\longrightarrow \mathbb {R}^n\) $\mathcal{L}:\mathcal{D}\times \mathbb{R}\times {\mathbb{R}}^n⟶{\mathbb{R}}^n$ , and \(\mathcal {L}_0:\mathcal {D}\times \mathbb {R}\times \mathbb {R}^n\longrightarrow \mathbb {R}\) ${\mathcal{L}}_0:\mathcal{D}\times \mathbb{R}\times {\mathbb{R}}^n⟶\mathbb{R}$ are Carathéodory applications that verified some conditions, and \(h_0\in L^1(\mathcal {D})~\) $h_0\in L^1(\mathcal{D})$ and \(h_j\in L^{p'}(\mathcal {D},\vartheta _{1}^{1-p'})(j=1,\ldots ,n)\) $h_j\in L^{p^{\prime }}(\mathcal{D},{\vartheta }_1^{1-p^{\prime }})(j=1,\dots ,n)$ .