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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n \in ({\frac{5}{2}},5]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月8日

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  • Giulio Ciraolo,
  • Camilla Chiara Polvara

摘要

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 \(\mathbb {R}^d\) R d , with \(d\ge 2\) d ≥ 2 . We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension \(n \in ({\frac{5}{2}},5]\) n ∈ ( 5 2 , 5 ] .