Maximizers for the variational problems associated with Sobolev type inequalities under constraints

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}.