Bounded solutions for nonlinear elliptic equations with degenerate coercivity and data in an $L\log L$

Abstract

In this paper, we prove $L^\infty$-regularity for solutions of some nonlinear elliptic equations with degenerate coercivity whose prototype is $$ \left\{\begin{array}{lll} {\rm-div}({\frac{1}{(1+|u|)^{\theta(p-1)}}}|\nabla u|^{p-2}{\nabla u})=f&{\rm in}&\Omega, \\ u=0&{\rm on}& \partial{\Omega}, \end{array} \right. $$ where $\Omega$ is a bounded open set in ${\rm \mathbb{R}^N}$, $N\geq 2$, $1<p<N$, $\theta$ is a real such that $0\leq\theta\leq1$ and $f\in L^{\frac{N}{p}}log^{\alpha}L$ with some $\alpha>0.$