The asymptotic behaviour of solutions of second-order quasilinear elliptic partial differential equations defined on unbounded domains in Rn contained in strips (when n = 2) or slabs (when n > 2) is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data have appropriate asymptotic behaviour at infinity. We prove Phragmèn–Lindelöf theorems for large classes of elliptic operators, including uniformly elliptic operators and operators with well-defined genre, establish exponential decay estimates for uniformly elliptic operators when the Dirichlet boundary data vanish outside a compact set, establish the uniqueness of solutions, and give examples of solutions for non-uniformly elliptic operators which decay but do not decay exponentially. Our principal theorems are proven using special barrier functions; these barriers are constructed by considering an operator associated to our original operator.