  We prove that the Dirichlet problem for the Lane-Emden equation in a
half-space has no positive solution which is monotone in the normal direction.
As a consequence, this problem does not admit any positive classical solution
which is bounded on finite strips. This question has a long history and our
result solves a long-standing open problem. Such a nonexistence result was
previously available only for bounded solutions, or under a restriction on the
power in the nonlinearity. The result extends to general convex nonlinearities.
