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 $\Omega\subset \mathbb R^N$ $(N\geq 2)$, 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.