Unilateral Elliptic Problems with $L 1$−data in Anisotropic Weighted Sobolev Spaces

Abstract

In this note, we are interested in some results on the existence of entropy solution for quasilinear anisotropic unilateral elliptic problem of the type:

\begin{equation}\label{prbm1.0.0}
\left\{\begin{array}{ll}
-\sum_{i=1}^{N} \partial^{i} a_{i}(x, u, \nabla u)+\Phi(x, u, \nabla u)+H(x, u, \nabla u)=f \quad\text { in } \Omega \\\\
u \geq \varphi \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad \text { a.e. in } \Omega,
\end{array}\right.
\end{equation}
where $ f \in L^{1}(\Omega), $ The nonlinear terms $ \Phi(x, s, \nabla u) $ satisfy the sign and growth conditions, and $ H(x, s, \nabla u) $ verifies only the growth conditions.