  We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on
a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions
on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator
of order $2m$ whose principle part is of the form $\big(-\sum_{i,j=1}^N
a_{ij}(x) \frac{\partial^2}{\partial x_i\partial x_j}\big)^m$. We assume that
$f$ is superlinear at the origin and satisfies $\lim
\limits_{s\to\infty}\frac{f(x,s)}{s^q}=h(x)$, $\lim
\limits_{s\to-\infty}\frac{f(x,s)}{|s|^q}=k(x)$, where $h,k\in
C(\overline{\Omega})$ are positive functions and $q>1$ is subcritical. By
combining degree theory with new and recently established a priori estimates,
we prove the existence of a nontrivial solution.
