Critical dimensions for polyharmonic operators: The Pucci-Serrin conjecture for solutions of bounded energy

TL;DR

This paper proves a conjecture regarding critical dimensions for polyharmonic operators with bounded energy, using Green's function analysis of operators with near Hardy potential.

Contribution

It introduces a novel approach analyzing Green's functions to establish the Pucci-Serrin conjecture for polyharmonic operators.

Findings

Confirmed the Pucci-Serrin conjecture in specific critical dimensions.

Developed a method based on Green's function analysis for polyharmonic operators.

Extended understanding of the behavior of solutions with bounded energy.

Abstract

We prove a Pucci-Serrin conjecture on critical dimensions under a uniform bound on the energy. The method is based on the analysis of the Green's function of polyharmonic operators with "almost" Hardy potential.