We prove a Phragmèn-Lindelöf theorem which yields the behavior at infinity of bounded solutions of Dirichlet problems for non-hyperbolic (e.g., elliptic, parabolic) quasilinear second-order partial differential equations in terms of particular solutions of appropriate ordinary differential equations.

We prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space (R", go) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at ~.

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The existence of solutions of the Dirichlet problems for the prescribed mean-curvature equation on some unbounded domains in R n (n 2) is proved. The results are proved using a modified version of the Perron method, where a subsolution is a solution to the minimal surface equation, while a supersolution is not constructed; instead, the role played by a supersolution is replaced by the estimates on the uniform bounds on the liftings of subfunctions on compact sets.