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Bounds for the best constant in an improved Hardy-Sobolev inequality. (English) Zbl 1095.26008
Let $$\Omega$$ be a bounded domain in $$\mathbb R^n$$ $$(n \geq 2)$$ with $$0 \in \Omega$$. In [C. N. Adimurthi and M. Ramaswamy, Proc. Am. Math. Soc. 130, No. 2, 489–505 (2002; Zbl 0987.35049)], an improved Hardy-Sobolev inequality was established: for all $$u \in W_0^{1,p} (\Omega)$$,
$\int_\Omega | \nabla u |^p \,dx \geq ({n-p \over p})^p \int_\Omega {| u |^p \over | x |^p} \,dx + C \int_\Omega {| u |^p \over | x |^p} [\log {R \over | x | }]^{-2} \,dx,$ where $$1 < p < n$$, $$R \geq e^{2/p} \sup_\Omega | x |$$ and $$C$$ is a positive constant.
Define $$C = C(n,p,R,\Omega) = \inf_{0 \not = u \in W_0^{1,p} (\Omega)} Q_{\Omega, R} (u)$$ with $Q_{\Omega, R} (u) = \frac{\int_\Omega | \nabla u |^p \,dx - ({n-p \over p})^p \int_\Omega {| u |^p \over | x |^p} \,dx}{\int_\Omega {| u(x) |^p \over | x |^p} (\log {R \over | x |} )^{-2} \,dx}$ to be the best constant of the inequality. In this article, the author shows with an interesting proof that in the first place $$C(n,p,R,\Omega) = C(n,p)$$, that is, the best constant is independent of $$R$$ and $$\Omega$$. Then for $$2 \leq p < n$$, it is shown that $${p-1 \over p^2} ({n-p \over p})^{p-2} \leq C(n, p) \leq {p-1 \over 2} ({n-p \over p})^{p-2}$$.

##### MSC:
 26D10 Inequalities involving derivatives and differential and integral operators 35J20 Variational methods for second-order elliptic equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
Hardy-Sobolev inequality; best constant
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##### References:
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