Explicit estimates for solutions to higher order elliptic PDEs via Morse index

TL;DR

This paper derives explicit $L^{ty}$ and $L^{p}$ estimates for solutions of polyharmonic elliptic equations using Morse index, marking a novel contribution in the analysis of higher order PDEs.

Contribution

It provides the first explicit $L^{ty}$ and $L^{p}$ estimates for polyharmonic elliptic problems based on Morse index.

Findings

Established explicit $L^{ty}$ estimates for polyharmonic solutions

Derived $L^{p}$ bounds related to Morse index

First known estimates of this kind for higher order elliptic PDEs

Abstract

In this paper, we establish $L^{\infty}$ and $L^{p}$ estimates for solutions of some polyharmonic elliptic equations via the Morse index. As far as we know, it seems to be the first time that such explicit estimates are obtained for polyharmonic problems.