On solutions of second order elliptic equations in cylindrical domains

In a semi-infinite cylinder, we consider a second order elliptic equation with a lower order term. On the lateral boundary of the cylinder we impose the homogeneous Neumann condition. We show that each bounded solution tends to a constant at infinity and once the lower order term does not decay too fast, this constant vanishes. We establish that for a sufficiently fast decay of the lower order term, we have a trichotomy of the solutions as for the equation without the lower order term: the solution tends to a general non-zero constant or grows linearly or grows exponentially. The decay conditions for the lower order term are formulated in an integral form.