Interpolation inequality and some applications

TL;DR

This paper introduces a new interpolation inequality to derive explicit universal estimates for solutions of polyharmonic equations with finite Morse index, providing insights into their growth and regularity.

Contribution

It presents a direct proof of universal estimates using a novel interpolation inequality, improving understanding of solution behavior under large growth conditions.

Findings

Universal constant grows as a power function of Morse index.

Provides local $L^p$-$W^{2r,p}$ estimates for solutions.

Offers a new proof technique distinct from previous works.

Abstract

We investigate {\bf explicit} universal estimate of finite Morse index solutions to polyharmonic equations. \,Differently to previous works \cite{BL2, DDF, fa, H1}, propose here a direct proof using a new interpolation inequality and a delicate boot-strap argument under large superlinear and subcritical growth conditions to show that the universal constant grows as a power function of the Morse index.\, Also, our interpolation inequality allows us to provide local $L^p$-$W^{2r,p}$ estimate.