Existence and non-existence of maximizers for a class weighted variational problems under equivalent constraints

In this paper, we study the existence and non-existence of maximizers for variational problems associated with Caffarelli-Kohn-Nirenberg-type inequalities under the equivalent constraints both in the subcritical case and critical case. Furthermore, if the maximizer exists, we can derive the concrete form of the maximizers. Our method is based on establishing a useful link between the attainability of the supremum in our variational problems and the attainability of the supremum of some special functions defined on (0,infinity).