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• W4307380903 abstract "We study the sharp constant for the embedding of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W 0 Superscript 1 comma p Baseline left-parenthesis normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mn>0</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W^{1,p}_0(Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript q Baseline left-parenthesis normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^q(Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 greater-than p greater-than q> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>2>p>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that for smooth connected sets, when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q greater-than p> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>q>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sufficiently close to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p minus> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>Laplace–type equations by L. Damascelli and B. Sciunzi." @default.
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• W4307380903 date "2023-02-09" @default.
• W4307380903 modified "2023-09-30" @default.
• W4307380903 title "Uniqueness of extremals for some sharp Poincaré-Sobolev constants" @default.
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