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From fractional Lane-Emden-Serrin equation -- existence, multiplicity and local behaviors via classical ODE -- to fractional Yamabe metrics with singularity of "maximal" dimension

Hardy Chan, Azahara DelaTorre

Abstract

Point singularities of solutions to the classical Lane-Emden-Serrin equation have a polyhomogeneous asymptotic expansion whose logarithmic corrections are determined by a first order ODE. Surprisingly, we are able to discover such an ODE for the fractional Lane-Emden-Serrin equation, and therefore give a short classification for the precise local behavior of its solutions up to the second order involving a double logarithm. This seems to be the first time that a nonlocal equation is associated to a genuinely local ODE in one dimension. New non-existence, existence and multiplicity results for the corresponding Dirichlet problem are also discussed. Moreover, we construct complete $s$-fractional Yamabe metrics in $\mathbb{R}^n$ which are singular along a smooth submanifold of dimension $(n-2s)/2$, via direct integral asymptotic analysis with global geometric weights. This covers the missing borderline case as suggested by the deep work of Schoen and Yau. While such dimension is maximal in the class of distributional solutions, we conjecture the existence of complete metrics, understood in a suitably generalized sense, with prescribed singularities of strictly higher dimensions.

From fractional Lane-Emden-Serrin equation -- existence, multiplicity and local behaviors via classical ODE -- to fractional Yamabe metrics with singularity of "maximal" dimension

Abstract

Point singularities of solutions to the classical Lane-Emden-Serrin equation have a polyhomogeneous asymptotic expansion whose logarithmic corrections are determined by a first order ODE. Surprisingly, we are able to discover such an ODE for the fractional Lane-Emden-Serrin equation, and therefore give a short classification for the precise local behavior of its solutions up to the second order involving a double logarithm. This seems to be the first time that a nonlocal equation is associated to a genuinely local ODE in one dimension. New non-existence, existence and multiplicity results for the corresponding Dirichlet problem are also discussed. Moreover, we construct complete -fractional Yamabe metrics in which are singular along a smooth submanifold of dimension , via direct integral asymptotic analysis with global geometric weights. This covers the missing borderline case as suggested by the deep work of Schoen and Yau. While such dimension is maximal in the class of distributional solutions, we conjecture the existence of complete metrics, understood in a suitably generalized sense, with prescribed singularities of strictly higher dimensions.

Paper Structure

This paper contains 61 sections, 46 theorems, 295 equations, 1 table.

Table of Contents

  1. Introduction
  2. Overview
  3. Organization
  4. Notations
  5. Remark on the title
  6. Added comment
  7. The Lane--Emden--Serrin equation
  8. Local singular behavior for classical Lane--Emden equations
  9. Local singular behavior for fractional Lane--Emden equations
  10. Non-existence of global solutions
  11. Geometric motivation
  12. Singular Yamabe problem
  13. Singular fractional Yamabe problem
  14. Arbitrarily singular incomplete metrics
  15. Dimensional restriction for complete metrics
  16. ...and 46 more sections

Key Result

Theorem 4.1

Let $u\in C^2(B_1\setminus\left\{0\right\})$ be a solution of eq:loc-beh--eq:g-finite. Then, either $u$ has a removable singularity at the origin, or where $c_0>0$, $c_1<0$ are constants given in eq:c0-explicit and rmk:c1.

Theorems & Definitions (94)

  • Remark 3.1
  • Conjecture 3.1
  • Remark 3.2
  • Theorem 4.1: Exact local behavior
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 4.2: Liouville theorem for exterior solutions, integral version
  • Theorem 4.3: Liouville theorem for exterior solutions, integro-differential version
  • Theorem 4.4
  • ...and 84 more