Picone's Identity for $p$-biharmonic operator and Its Applications

TL;DR

This paper establishes a nonlinear Picone's identity for the p-biharmonic operator and applies it to derive properties like Morse index, Hardy inequalities, and eigenvalue monotonicity in related boundary value problems.

Contribution

It introduces a nonlinear Picone's identity for p-biharmonic operators and demonstrates its applications in spectral theory and inequalities.

Findings

Morse index of zero solution is zero

Established Hardy type inequality

Proved strict monotonicity of principal eigenvalue

Abstract

In this article we prove the nonlinear analogue of Picone's identity for $p-$biharmonic operator. As an application of our result we show that the Morse index of the zero solution to a $p-$biharmonic boundary value problem is $0$. We also prove a Hardy type inequality and Sturmian comparison principle. We also show the strict monotonicity of the principle eigenvalue and linear relationship between the solutions of a system of singular $p$-biharmonic system.