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

Frédéric Robert

TL;DR

This work resolves the Pucci-Serrin conjecture on critical dimensions for radial solutions of the polyharmonic equation in a unit ball under a uniform energy bound, in the range $2k<n<4k$. The authors develop a uniform, concentration-based analysis that combines sharp Green's function estimates for an operator with an almost Hardy potential, a multi-bubble blow-up framework, and a Pohozaev-Pucci-Serrin-type identity to derive a contradiction unless the energy is controlled. A key technical advance is the construction and precise asymptotics of Green's functions for $\Delta^k+h-V$ under Hardy-type perturbations, together with a robust regularity theory in the spirit of Agmon-Douglis-Nirenberg. The results yield rigorous control of potential bubbles, provide integral and pointwise bounds on Green's functions, and deliver a versatile framework applicable to higher-order elliptic problems with critical nonlinearities.

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.

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

TL;DR

This work resolves the Pucci-Serrin conjecture on critical dimensions for radial solutions of the polyharmonic equation in a unit ball under a uniform energy bound, in the range . The authors develop a uniform, concentration-based analysis that combines sharp Green's function estimates for an operator with an almost Hardy potential, a multi-bubble blow-up framework, and a Pohozaev-Pucci-Serrin-type identity to derive a contradiction unless the energy is controlled. A key technical advance is the construction and precise asymptotics of Green's functions for under Hardy-type perturbations, together with a robust regularity theory in the spirit of Agmon-Douglis-Nirenberg. The results yield rigorous control of potential bubbles, provide integral and pointwise bounds on Green's functions, and deliver a versatile framework applicable to higher-order elliptic problems with critical nonlinearities.

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.
Paper Structure (13 sections, 23 theorems, 208 equations)

This paper contains 13 sections, 23 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Preliminary analysis
  3. Sobolev spaces and inequalities
  4. Sharp analysis at the furthest scale
  5. Conclusion via the Pohozaev-Pucci-Serrin identity
  6. Green's function for an "almost" Hardy operator
  7. Construction of the Green's function
  8. Asymptotics for the Green's function close to the singularity
  9. Asymptotics for the Green's function far from the singularity
  10. Proof of Theorem \ref{['APP:GREEN:th:Green:pointwise:BIS']}
  11. The regularity Lemma
  12. Green's function for elliptic operators with bounded coefficients
  13. Regularity theorems

Key Result

Theorem 1.1

Let $B$ be the unit ball of $\mathbb{R}^n$ and let $k\in\mathbb{N}$ be such that $n>2k\geq 2$. Assume that Then, for any $M>0$, there exists $\lambda_0(n,k,M)>0$ such that for all $0<\lambda<\lambda_0(n,k,M)$, any radial solution to eq:1 satisfying that $\Vert u\Vert_{2^\star}\leq M$ is identically null.

Theorems & Definitions (37)

  • Conjecture 1.1
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 27 more