Existence of extremal functions for a Hardy-Sobolev inequality
The best constant and the extremal functions for the Improved Hardy-Sobolev inequality are obtained. Also, the author proved that under the transformation \(v(r)=u(t)\), \(t=(-\log (\frac rR))^{-\frac{1}{N-2}}\) the following inequalities are equivalent: (i) \(\int_0^Rr|v'|^2dr \geq c(\int_0^Rr^{-1}(-\log (\frac rR))^{-\frac{2(N-1)}{N-2}} |v|^{\frac{2N}{N-2}}dr)^{\frac{N-2}{N}}\), \(v\in C_0^{\infty}(0,R)\), (ii) \(\int_{\mathbb{R}^N}|\nabla u|^2dx \geq S(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-2}}dx)^{\frac{N-2}{N}}\) (Sobolev inequality).
scientific article; zbMATH DE number 5731611
scientific article