A Cauchy-Liouville type theorem is a statement that under appropriate circumstances an entire solution (a solution defined over ℝ
) of an elliptic equation must be constant.
For the Laplace equation in particular, it is enough that a solution
should be bounded, or even, at a minimum, that
|) as |
| → ∞. For quasilinear equations, and even for semilinear equations of the form Δ
) = 0,
, (8.1.1) the same question is more delicate than might at first be expected, since a number of different kinds of behavior can be seen even for relatively simple examples.