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Bol's type inequality for singular metrics and its application to prescribing $Q$-curvature problems

Mrityunjoy Ghosh, Ali Hyder

TL;DR

This work establishes higher order Bol's inequalities for radial normal solutions of singular ${\mathcal{Q}}$-curvature equations on ${\mathbb R}^n$ and uses them together with decay estimates and Pohozaev identities to obtain sharp existence criteria and uniform total curvature bounds. The authors derive necessary and sufficient conditions on the total curvature $\Lambda_*$ for radial normal solutions to exist, and show that under suitable growth conditions on the prescribed curvature $Q$ one can uniformly bound $\Lambda_*$. They develop fixed point arguments to construct solutions in critical and supercritical regimes and analyze the asymptotics of the total curvature as the solution mass varies. The results extend the classical Bol's inequality to singular settings and provide robust compactness tools for prescribing ${\mathcal{Q}}$-curvature in noncompact geometries.

Abstract

In this article, we study higher-order Bol's inequality for radial normal solutions to a singular Liouville equation. By applying these inequalities along with compactness arguments, we derive necessary and sufficient conditions for the existence of radial normal solutions to a singular $Q$-curvature problem. Moreover, under suitable assumptions on the $Q$-curvature, we obtain uniform bounds on the total $Q$-curvature.

Bol's type inequality for singular metrics and its application to prescribing $Q$-curvature problems

TL;DR

This work establishes higher order Bol's inequalities for radial normal solutions of singular -curvature equations on and uses them together with decay estimates and Pohozaev identities to obtain sharp existence criteria and uniform total curvature bounds. The authors derive necessary and sufficient conditions on the total curvature for radial normal solutions to exist, and show that under suitable growth conditions on the prescribed curvature one can uniformly bound . They develop fixed point arguments to construct solutions in critical and supercritical regimes and analyze the asymptotics of the total curvature as the solution mass varies. The results extend the classical Bol's inequality to singular settings and provide robust compactness tools for prescribing -curvature in noncompact geometries.

Abstract

In this article, we study higher-order Bol's inequality for radial normal solutions to a singular Liouville equation. By applying these inequalities along with compactness arguments, we derive necessary and sufficient conditions for the existence of radial normal solutions to a singular -curvature problem. Moreover, under suitable assumptions on the -curvature, we obtain uniform bounds on the total -curvature.
Paper Structure (8 sections, 19 theorems, 182 equations)

This paper contains 8 sections, 19 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Decay estimates and Pohozaev identities
  3. Decay properties
  4. Pohozaev-type identities
  5. Proofs of Theorems \ref{['Bols_up']} and \ref{['Bols_low']}
  6. Proof of Theorem \ref{['existence_coro']}
  7. Proof of Theorem \ref{['f_existence']}
  8. Proof of Theorem \ref{['total-curvature']}

Key Result

Theorem 1.2

Let $u$ be a radial normal solution to Q-equation for some $\alpha>-1$ and $Q\in L^\infty_{\rm loc}({\mathbb R}^n)$. Then if $Q\leq 1,$ we have Moreover, the equality holds if and only if $Q\equiv 1.$

Theorems & Definitions (41)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 31 more