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

ABSTRACT

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 $(0,\mathrm{\infty })$.

KEYWORDS:

• Weighted variational problems
• existence and non-existence
• Caffarelli–Kohn–Nirenberg inequalities

AMS SUBJECT CLASSIFICATIONS:

• 46E35
• 26D10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by the Natural Science Foundation of China [12071185,12061010,11971202], Natural Science Foundation of Jiangsu Province [BK20160483], Outstanding Young Foundation of Jiangsu Province [BK20200042] and Jiangsu University Foundation Grant [16JDG043].