  We establish uniform a-priori estimates for solutions of the semilinear
Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m
u=h(x,u)\quad&\mbox{in }\Omega,\\
u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega,
\end{cases} \end{equation} where $h$ is a positive superlinear and subcritical
nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when
$\Omega$ is a ball or, provided an energy control on solutions is prescribed,
when $\Omega$ is a smooth bounded domain. The analogue problem with Navier
boundary conditions is also studied. Finally, as a consequence of our results,
existence of a positive solution is shown by degree theory.
