Advances in Differential Equations

A family of sharp inequalities for Sobolev functions

Pedro M. Girão

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 8, Number 6 (2003), 641-671.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355926830

Mathematical Reviews number (MathSciNet)
MR1969650

Zentralblatt MATH identifier
1049.46017

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations

Citation

Girão, Pedro M. A family of sharp inequalities for Sobolev functions. Adv. Differential Equations 8 (2003), no. 6, 641--671. https://projecteuclid.org/euclid.ade/1355926830.


Export citation