given the minimization problem:
$inf \ \frac{\int_{\Omega} |\nabla u|^p }{ \int_{\Omega} \frac{|u|^p}{|x|^p} } ,\ \ p>1$
infimum taken on all smooth functions with compact support in the punctured space: $R^n-{0}$
how can I show that it is enough to take the infimum just on all radially symmetric
functions in the punctured space?
note :we have the classical inequality :
$\int_{R^n}|\nabla u|^p>= \left (\frac{n-p}{p} \right )^p\int_{R^n} \frac{|u|^p}{|x|^p}$
for all smooth functions with compact support