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On the classification of extremals of Caffarelli-Kohn-Nirenberg inequalities

Giulio Ciraolo, Camilla Chiara Polvara

Abstract

We consider a family of critical elliptic equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities, possibly in convex cones in $\mathbb{R}^d$, with $d\geq 2$. We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension $n \in (\frac{3}{2},5]$.

On the classification of extremals of Caffarelli-Kohn-Nirenberg inequalities

Abstract

We consider a family of critical elliptic equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities, possibly in convex cones in , with . We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension .

Paper Structure

This paper contains 6 sections, 12 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. The conformal formulation
  3. The p-function
  4. Proof of Theorem \ref{['thm_main']}
  5. Classification in cones
  6. A differential identity in Riemannian Manifolds

Key Result

Theorem 1.1

Let $u \in \mathcal{D}^{a,b}_{loc} (\mathbb R^d)$ be a positive solution to eq_CKN and let $\alpha$ and $n$ be given by alpha_def and n_def, respectively. If then $u$ is radially symmetric and it is given by u_radial (up to a translation in the case $a=b=0$).

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 14 more