  We study a mixed boundary value problem for the quasilinear elliptic equation
$\mathop{\rm div}\mathcal{A}(x,\nabla u(x))=0$ in an open infinite circular
half-cylinder with prescribed continuous Dirichlet data on a part of the
boundary and zero conormal derivative on the rest. We prove the existence and
uniqueness of bounded weak solutions to the mixed problem and characterize the
regularity of the point at infinity in terms of \p-capacities. For solutions
with only Neumann data near the point at infinity we show that they behave in
exactly one of three possible ways, similar to the alternatives in the
Phragm\'en--Lindel\"of principle.
