Extremal values of the (fractional) Weinstein functional on the hyperbolic space

Abstract

We make a study of Weinstein functionals, first defined in ~\cite{W}, on the hyperbolic space ${\mathbb{H}}^n$. We are primarily interested in the existence of Weinstein functional maximisers, or, in other words, existence of extremal functions for the best constant of the Gagliardo-Nirenberg inequality. The main result is that the maximum value of the Weinstein functional on ${\mathbb{H}}^n$ is the same as that on ${\mathbb{R}}^n$ and the related fact that the maximum value of the Weinstein functional is not attained on ${\mathbb{H}}^n$, when maximisation is done in the Sobolev space $H^1({\mathbb{H}}^n)$. This proves a conjecture made in ~\cite{CMMT} and also answers questions raised in several other papers (see, for example, ~\cite{B}). We also prove that a corresponding version of the conjecture will hold for the Weinstein functional with the fractional Laplacian as well.

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