Abstract
We study Weinstein functionals, first defined in [33], mainly on the hyperbolic space
ℍ
n
{\mathbb{H}^{n}}
.
We are primarily interested in the existence of Weinstein functional maximizers or, in other words, existence of extremal functions for the best constant of the Gagliardo–Nirenberg inequality.
The main result is that the supremum of the Weinstein functional on
ℍ
n
{\mathbb{H}^{n}}
is the same as that on
ℝ
n
{\mathbb{R}^{n}}
and the related fact that the said supremum is not attained on
ℍ
n
{\mathbb{H}^{n}}
, when functions are chosen from the Sobolev space
H
1
(
ℍ
n
)
{H^{1}(\mathbb{H}^{n})}
.
This proves a conjecture made in [8] (see also [3]).
We also prove an analogous version of the conjecture for the Weinstein functional defined with the fractional Laplacian.</jats:p
