Compactness of solutions to higher order elliptic equations

TL;DR

This paper proves the compactness of solutions to higher order critical elliptic equations with non-degenerate zero potentials and establishes a Laplacian vanishing rate at blow-up points, advancing understanding of elliptic PDE behavior.

Contribution

It introduces blow-up analysis for local integral equations to demonstrate solution compactness and confirms a conjecture relating to the vanishing rate of potentials at blow-up points.

Findings

Solutions are compact under non-degenerate zero potentials.

Laplacian of potentials vanishes at blow-up points at a specific rate.

Supports Schoen's Weyl tensor vanishing conjecture for the Yamabe problem.

Abstract

We use blow up analysis for local integral equations to prove compactness of solutions to higher order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.