Boundedness of solutions to singular anisotropic elliptic equations

Abstract

We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$-{\Delta }_{\overrightarrow{p}}u+{\Phi }_0(u,\nabla u)=\Psi (u,\nabla u)+f$$ on a bounded open subset $\Omega \subset {\mathbb{R}}^N$ $(N\ge 2)$, where ${\Delta }_{\overrightarrow{p}}u={\sum }_{j=1}^N{\partial }_j(|{\partial }_{j}u{|}^{p_j-2}{\partial }_{j}u)$ and ${\Phi }_0(u,\nabla u)=\left({\mathfrak{a}}_0+{\sum }_{j=1}^N{\mathfrak{a}}_j|{\partial }_{j}u{|}^{p_j}\right)|u{|}^{m-2}u$, with ${\mathfrak{a}}_0>0$, $m,p_j>1$, ${\mathfrak{a}}_j\ge 0$ for $1\le j\le N$ and $N/p={\sum }_{k=1}^N(1/p_k)>1$. We assume that $f\in L^r(\Omega )$ with $r>N/p$. The feature of this study is the inclusion of a possibly singular gradient-dependent term $\Psi (u,\nabla u)={\sum }_{j=1}^N|u{|}^{{\theta }_j-2}u|{\partial }_{j}u{|}^{q_j}$, where ${\theta }_j>0$ and $0\le q_j<p_j$ for $1\le j\le N$. The existence of such weak solutions is contained in a recent paper by the authors.