  In this paper, we study the existence of uniform a priori estimates for
positive solutions to Navier problems of higher order Lane-Emden equations
\begin{equation*}
  (-\Delta)^{m}u(x)=u^{p}(x), \qquad \,\, x\in\Omega \end{equation*} for all
large exponents $p$, where $\Omega\subset\mathbb{R}^{n}$ is a star-shaped or
strictly convex bounded domain with $C^{2m-2}$ boundary, $n\geq4$ and $2\leq
m\leq\frac{n}{2}$. Our results extend those of previous authors for second
order $m=1$ to general higher order cases $m\geq2$.
