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}\label{0-0} (-\Delta)^{m}u(x)=u^{p}(x), \quad 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 $m\geq2$.