Existence and uniqueness of weak solution in weighted Sobolev spaces for a class of nonlinear degenerate elliptic problems with measure data

Document Type : Research Paper

Authors

Laboratory LMACS, FST of Beni Mellal, BP 523, 23000, Sultan Moulay Slimane University, Morocco

Abstract

In this paper, we study the existence and uniqueness of weak solution to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic problems
\begin{equation*}
-{\rm{div}}\Big[ \omega_{1}\mathcal{A}(x,\nabla u)+\nu_{2}\mathcal{B}(x,u,\nabla u)\Big]+ \nu_{1}\mathcal{C}(x,u)+ \omega_{2}\vert u\vert^{p-2}u=f-{\rm{div}}F,
\end{equation*}
where $1 < p < \infty$, $\omega_{1}$, $\nu_{2}$, $\nu_{1}$ and $\omega_{2}$ are $A_p$-weight functions, and $\mathcal{A}:\Omega\times \mathbb{R}^n\longrightarrow\mathbb{R}^n$, $\mathcal{B}:\Omega\times\mathbb{R}\times \mathbb{R}^n\longrightarrow\mathbb{R}^n$, $\mathcal{C}:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ are Carat'eodory functions that satisfy some conditions and the right-hand side term $f-{\rm{div}}F$ belongs to $L^{p'}(\Omega,\omega_{2}^{1-p'})+\prod\limits_{j=1}^{n}L^{p'}(\Omega,\omega_{1}^{1-p'})$. We will use the Browder-Minty Theorem and the weighted Sobolev spaces theory to prove the existence and uniqueness of weak solution in the weighted Sobolev space $W^{1,p}_ 0(\Omega,\omega_1,\omega_{2})$.

Keywords