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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
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