We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$ with $p \in (1,\infty$). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.

翻译：本文证明了凸区域上具有奇异强迫项的特定线性和非线性椭圆边值问题在加权Sobolev空间中的适定性。假设权重属于Muckenhoupt类$A_p$，其中$p \in (1,\infty)$。针对上述非线性椭圆边值问题，我们提出并分析了一种收敛的有限元离散格式。作为关键工具性结果，我们证明了特定线性问题离散格式在加权空间中的适定性。