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


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


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
-{\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,
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 $W^{1,p}_ 0(\Omega,\omega_1,\omega_{2})$.


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Volume 13, Issue 1
March 2022
Pages 2635-2653
  • Receive Date: 05 June 2021
  • Revise Date: 24 August 2021
  • Accept Date: 30 October 2021