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 $\Omega\subset \mathbb R^N$ $(N\geq 2)$, 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 $\mathfrak{ a}_0>0$, $m,p_j>1$, $\mathfrak{ a}_j\geq 0$ for $1\leq j\leq N$ 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 $0\leq q_j<p_j$ for $1\leq j\leq N$. The existence of such weak solutions is contained in a recent paper by the authors.
Categories: math.AP