Anisotropic Elliptic Nonlinear Obstacle Problem with Weighted Variable Exponent

Abstract

In this paper, we are concerned with a show the existence of a entropy solution to the obstacle problem associated with the equation of the type :

$\begin{cases} Au+g(x,u,∇u) = f  & {\rm in}  & Ω \\ u=0 & {\rm on} & ∂Ω \end{cases}$

where $\Omega$ is a bounded open subset of $\;\mathbb{R}^{N}$, $N\geq 2$, $A\,$ is an operator of Leray-Lions type acting from $\; W_{0}^{1,\overrightarrow{p}(.)} (\Omega,\ \overrightarrow{w}(.))\;$ into its dual $\;  W_{0}^{-1,\overrightarrow{p}'(.)} (\Omega,\ \overrightarrow{w}^*(.))$ and $\,L^1\,-\,$deta. The nonlinear term $\;g\,$: $\Omega\times \mathbb{R}\times \mathbb{R}^{N}\longrightarrow \mathbb{R} $ satisfying only some growth condition.