Boundedness of solutions to anisotropic second order elliptic equations in unbounded domains

Abstract

In the paper we study a class of anisotropic second order elliptic equations represented by the model equation $$ \sum_{\alpha=1}^n(|u_{x_\alpha}|^{p_\alpha-2}u_{x_\alpha})_{x_\alpha}=\sum_{\alpha=1}^n\left(\Phi_\alpha(\mathbf x)\right)_{x_\alpha},\quad p_n\geq\ldots\geq p_1>1. $$ We prove the boundedness of solutions to the homogeneous Dirichlet problem in unbounded domains located along one of the coordinate axes. We also establish an estimate for the solutions to the considered equations with a compactly supported right hand side that ensures a power decay of the solutions at infinity.