Uniformly nonlinear elliptic problems with obstacle

Abstract

We prove an existence result for solutions to a class of unilateral problems for the nonlinear elliptic equation whose prototype is $-div\left(|\nabla u{|}^{p-2}\nabla u\right)+b\left(x\right)|\nabla u{|}^{\lambda }=f-divF\text{in }\mathrm{\Omega }$, where $\mathrm{\Omega }$ is a bounded open set of ${ℝ}^N$, $N\ge 2$, $1<p<N$, $0\le \lambda \le p-1$, $b\left(x\right)$ belongs to the Lorentz space $L^{N,1}\left(\mathrm{\Omega }\right),f\in L^1\left(\mathrm{\Omega }\right)$ and $F\in {\left(L^{p^{\prime }}\left(\mathrm{\Omega }\right)\right)}^N$, $p^{\prime }=p∕\left(p-1\right)$.