This paper is devoted to the behaviour of solutions of \[ Lu:= \sum^n_{i,j=1} {\partial\over \partial x_i}\left (a_{ij}(x){\partial u\over \partial x_j}\right) =0 \] in unbounded domains having various structures: cones, cylinders, paraboloids, and others, with a nonlinear boundary condition \[ {\partial u\over\partial v}+b(x)|u|^{p-1}u=0 \quad\text{or}\quad {\partial u\over \partial v}- b(x)|u|^{p-1} u=0, \] where \(p>0\), \({\partial u\over \partial v}= \sum^n_{i,j=1} a_{ij}(x) {\partial u\over\partial x_j}\cos (n,x_i) \), \(b(x)\gtrless 0\), \(n\) is the outward pointing unit normal. The authors show that the behaviour of the solution essentially depends on the sign of the coefficient \(b(x)\). They prove a Phragmèn-Lindelöf type theorem, together with theorems on the existence of positive solutions and theorems on the lane of such solutions.