Boundedness of solutions to singular anisotropic elliptic equations

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