Advances in Differential Equations

A family of sharp inequalities for Sobolev functions

Pedro M. Girão

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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

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

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Girão, Pedro M. A family of sharp inequalities for Sobolev functions. Adv. Differential Equations 8 (2003), no. 6, 641--671.

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