Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A family of sharp inequalities for Sobolev functions   Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$,
${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left.
\int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}),
\nabla u\in L^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}}|u|^{2^*}=1 \right\}$ 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 $u\in H^1(\Omega)\setminus\{0\}$,
$$\frac{S}{2^{\frac 2N}}{|u|_{{2^*}}^2}\leq||u||^2+\alpha_{0}
\left(\frac{||u||}{|u|_{{2^*}}^{2^*/2}}\right)^s|u|_{q}^{qt},\qquad{(I)_{q}}$$
where the norms are over $\Omega$. Inequality $(I)_{{2^\flat}}$ is due to M.
Zhu.

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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an