Uniform a priori estimates for $n$-th order Lane-Emden system in $\mathbb{R}^{n}$ with $n\geq3$

TL;DR

This paper establishes uniform a priori estimates for positive solutions of higher-order Lane-Emden systems in arbitrary dimensions, extending previous results from lower dimensions and single equations to systems.

Contribution

The paper proves monotonicity of solutions and derives uniform estimates for higher-order Lane-Emden systems in dimensions n≥3, extending prior work from 2D and single equations to systems.

Findings

Extended uniform a priori estimates to higher dimensions n≥3

Proved monotonicity of solutions near the boundary

Established relationships between maxima of solutions and derivatives

Abstract

In this paper, we establish uniform a priori estimates for positive solutions to the (higher) critical order superlinear Lane-Emden system in bounded domains with Navier boundary conditions in arbitrary dimensions $n\geq3$. First, we prove the monotonicity of solutions for odd order (higher order fractional system) and even order system (integer order system) respectively along the inward normal direction near the boundary by the method of moving planes in local ways. Then we derive uniform a priori estimates by establishing the precise relationships between the maxima of $u$, $v$, $-\Delta u$ and $-\Delta v$ through Harnack inequality. Our results extended the uniform a priori estimates for critical order problems in [18, 19] from $n=2$ to higher dimensions $n\geq3$ and in [6, 8] from one single equation to system. With such a priori estimates, one will be able to obtain the existence of solutions via topological degree theory or continuation argument.