  We propose a new approach to study the existence and non-existence of
maximizers for the variational problems associated with Sobolev type
inequalities both in the subcritical case and critical case under the
equivalent constraints. The method is based on an useful link between the
attainability of the supremum in our variational problems and the attainablity
of the supremum of some special functions defined on $(0,\infty)$. Our approach
can be applied to the same problems related to the fractional Laplacian
operators. Our main results are new in the critical case and in the setting of
the fractional Laplacian operator which was left open in the work of Ishiwata
and Wadade \cite{Ishiwata,Ishiwata1}. In the subcritical case, our approach
provides a new and elementary proof of the results of Ishiwata and Wadade
\cite{Ishiwata,Ishiwata1}.
