Neumann problems in unbounded domains (Q1363305)

We consider equations of the form \[ Lu= \sum^n_{i,j=1}{\partial\over\partial x_i} \Biggl(a_{ij}(x){\partial u\over\partial x_j}\Biggr)= f\tag{1} \] in an unbounded domain \(\Omega\subset\mathbb{R}^n\) with a locally smooth boundary. Let \(u(x)\) be a solution to (1) that satisfies a homogeneous Neumann condition on the boundary. The aim of this work is to study the connection between the behavior of the solution \(u\) at \(\infty\) and the geometry of the domain \(\Omega\).