Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaces

Abstract

In this paper, we prove the existence and uniqueness of solution to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic equation

$$\begin{aligned} -\mathrm{{div}}\Big [ \omega _{1}{\mathcal {A}}(x,\nabla u)+\omega _{2}{\mathcal {B}}(x,u,\nabla u)\Big ]+ \omega _{3}b(x,u)+ \omega _{4}\vert u\vert ^{p-2}u=f(x), \end{aligned}$$

where $\omega _{1}$, $\omega _{2}$, $\omega _{3}$ and $\omega _{4}$ are weight functions, ${\mathcal {A}}:{\varOmega }\times {\mathbb {R}}^n\longrightarrow {\mathbb {R}}^n$, ${\mathcal {B}}:{\varOmega }\times {\mathbb {R}}\times {\mathbb {R}}^n\longrightarrow {\mathbb {R}}^n$, $b:{\varOmega }\times {\mathbb {R}}\longrightarrow {\mathbb {R}}$ are Caratéodory functions that satisfy some conditions and the right-hand side term f belongs to $L^1({\varOmega })$.