Uniform a priori estimates for positive solutions of higher order Lane-Emden equations in ${\mathbb{R}}^n$

Abstract

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*} (-Δ)^{m}u(x)=u^{p}(x), \qquad \,\, x\inΩ\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, $n\ge 4$ and $2\le m\le \frac{n}{2}$. Our results extend those of previous authors for second order $m=1$ to general higher order cases $m\ge 2$.