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

Mayukh Mukherjee

Published 2014-06-19, updated 2015-03-22Version 2

We make a study of Weinstein functionals, first defined in ~\cite{W}, on the hyperbolic space $\HH^n$. The main result is the fact that the maximum value of the Weinstein functional on $\HH^n$ is the same as that on $\RR^n$ and the related fact that the maximum value of the Weinstein functional is not attained on $\HH^n$, when maximisation is done in the Sobolev space $H^1(\HH^n)$. This proves a conjecture made in ~\cite{CMMT} and also answers questions raised in several other papers (see, for example, ~\cite{B}). Lastly, we prove that a corresponding version of the conjecture will hold for the Weinstein functional with the fractional Laplacian as well.