Existence and uniqueness result for a Navier problem involving Leray-Lions type operators in weighted Sobolev spaces

Abstract

We prove In this paper, the existence and uniqueness of solution for a Navier problem involving a general $p$-biharmonic operators associated with the degenerate nonlinear elliptic equation\\
\begin{equation*}
\Delta\Big[\phi(z)a(z,\Delta w)\Big]-{\rm{div}}\Big[ \vartheta_{1}(z)\mathcal{K}(z,\nabla w)\Big]+ \vartheta_2(z)\vert w\vert^{p-2}w=h(z),
\end{equation*}
where $\phi$,$\vartheta_1$ and $\vartheta_2$ are weight functions, $a:\overline{\mathcal{D}}\times \mathbb{R}^n\longrightarrow\mathbb{R}^n$, $\mathcal{K}:\mathcal{D}\times \mathbb{R}^n\longrightarrow\mathbb{R}^n$, are Carathéodory applications that verified some conditions, and $h$ belongs to $L^{p'}(\mathcal{D},\vartheta_{1}^{1-p'})$.