Trudinger type inequalities in 𝐑^𝐍 and their best exponents

1999/11/01 by Shinji Adachi, Kazunaga Tanaka · 14 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Advanced Harmonic Analysis Research

paper · pdf · doi:10.1090/s0002-9939-99-05180-1

Abstract

We study Trudinger type inequalities in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf R^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and their best exponents <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript upper N"> <mml:semantics> <mml:msub> <mml:mi> α </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">α _N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha element-of left-parenthesis 0 comma alpha Subscript upper N Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">α ∈ (0,α _N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript upper N Baseline equals upper N omega Subscript upper N minus 1 Superscript 1 slash left-parenthesis upper N minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:msubsup> <mml:mi> ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">α _N=Nω _N-1^1/(N-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript upper N minus 1"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">ω _N-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the surface area of the unit sphere in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf R^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ), there exists a constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript alpha Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">C_α &gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel integral Underscript bold upper R Superscript upper N Baseline Endscripts normal upper Phi Subscript upper N Baseline left-parenthesis alpha left-parenthesis StartFraction StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Over double-vertical-bar nabla u double-vertical-bar Subscript upper L Sub Superscript upper N Subscript left-parenthesis bold upper R Sub Superscript upper N Subscript right-parenthesis Baseline EndFraction right-parenthesis Superscript StartFraction upper N Over upper N minus 1 EndFraction Baseline right-parenthesis d x less-than-or-equal-to upper C Subscript alpha Baseline StartFraction double-vertical-bar u double-vertical-bar Subscript upper L Sub Superscript upper N Subscript left-parenthesis bold upper R Sub Superscript upper N Subscript rig

Cited by

• Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth
• Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth
• Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities
• The cone Moser-Trudinger inequalities and their applications
• Variational methods for degenerate Kirchhoff equations
• Scattering for the two-dimensional NLS with (full) exponential nonlinearity
• Concentration-compactness principle of singular Trudinger–Moser inequality involving N-Finsler–Laplacian operator
• Sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities and their extremal functions
• Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space
• Energy Critical NLS in two space dimensions
• On the thresholds of the critical singular anisotropic Trudinger–Moser inequality
• Concentration-compactness principle of singular Trudinger-Moser inequality involving N-Finsler-Laplacian operator
• Normalized solutions for Schrödinger equations with mixed dispersion and critical exponential growth in \mathbb R²
• A sharp Trudinger-Moser type inequality involving Lⁿ norm in the entire space ℝⁿ

Related

• On a theorem of Scott and Swarup
• On regular 𝐺-gradings
• The Stieltjes moments problem for rapidly decreasing functions
• A relation between pointwise convergence of functions and convergence of functionals
• Approximation on disks