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.