Quasilinear degenerate elliptic unilateral problems

Abstract

We will be concerned with the existence result of a degenerate elliptic unilateral problem of the form $Au+H\left(x,u,\nabla u\right)=f$, where $A$ is a Leray-Lions operator from $W^{1,p}\left(\Omega ,w\right)$ into its dual. On the nonlinear lower-order term $H\left(x,u,\nabla u\right)$, we assume that it is a Carathéodory function having natural growth with respect to $\left|\nabla u\right|$, but without assuming the sign condition. The right-hand side $f$ belongs to $L^1\left(\Omega \right)$.