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2003 A family of sharp inequalities for Sobolev functions
Pedro M. Girão
Adv. Differential Equations 8(6): 641-671 (2003). DOI: 10.57262/ade/1355926830

Abstract

Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb R^{N}$, ${{2^*}}=\frac{2N}{N-2}$, $a>0$, $S=\inf\big\{ \int_{\mathbb R^{N}}|\nabla u|^2 : u\in L^{{2^*}}(\mathbb R^{N}), \nabla u\in L^2(\mathbb R^{N}), \int_{\mathbb R^{N}}|u|^{{2^*}}$ $=1 \big\}$ and $||u||^2= |\nabla u|_{2}^2+a|u|_{2}^2$. We define ${{2^\flat}}=\frac{2N}{N-1}$, ${{2^\#}}=\frac{2(N-1)}{N-2}$ and consider $q$ such that ${{2^\flat}}\leq q\leq{{2^\#}}$. We also define $s=2-N+\frac{q}{{{2^*}}-q}$ and $t=\frac{2}{N-2}\cdot\frac{1}{{{2^*}}-q}$. We prove that there exists an $\alpha_{0}(q,a,\Omega)>0$ such that, for all $u\in H^1(\Omega)\setminus\{0\}$, $$ {\frac{S}{2^{\frac 2N}}} {{|u|_{{2^*}}^2}} \leq{||u||^2}+\alpha_{0} {\Big(\frac{{||u||}}{|u|_{{{2^*}}}^{{{2^*}}\!/2}}\Big)^{s}} {|u|_{q}^{qt}} , \tag*{(I)_{q}}$$ where the norms are over $\Omega$. Inequality $(I)_{{{2^\flat}}}$ is due to M. Zhu.

Citation

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Pedro M. Girão. "A family of sharp inequalities for Sobolev functions." Adv. Differential Equations 8 (6) 641 - 671, 2003. https://doi.org/10.57262/ade/1355926830

Information

Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1049.46017
MathSciNet: MR1969650
Digital Object Identifier: 10.57262/ade/1355926830

Subjects:
Primary: 46E35
Secondary: 35J20 , 35J60

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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Vol.8 • No. 6 • 2003
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