  We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on
a bounded smooth domain $\Omega$ in $R^N$ with Dirichlet boundary conditions.
The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that
for $s\to \pm\infty$ the nonlinearity $f(x,s)$ behaves like $|s|^q$ multiplied
by a continuous and positive function of $x$. Here the exponent $q$ is
subcritical, i.e., $q>1$ if $N<=2m$, $1<q<\frac{N+2m}{N-2m}$ if $N>2m$. We
prove a priori bounds, i.e, we show that the $L^\infty$-norm of every solution
$u$ is bounded by a constant independent of $u$. The solutions are allowed to
be sign-changing. The proof is done by a blow-up argument which relies on the
following new Liouville-type theorem on a half-space: if $u$ is a classical,
bounded, non-negative solution of $(-\Delta)^m u = u^q$ in a half-space with
Dirichlet boundary conditions and if $q>1$ is subcritical then $u$ vanishes
identically.
