  We consider the generalized Choquard equation describing trapped electron gas
in 3 dimensional case. The study of orbital stability of the energy minimizers
(known as ground states) depends essentially in the local uniqueness of these
minimizers. In equivalent way one can optimize the Gagliardo--Nirenberg
inequality subject to the constraint fixing the $L^2$ norm. The uniqueness of
the minimizers for the case $p=2$, i.e. for the case of Hartree--Choquard is
well known. The main difficulty for the case $p > 2$ is connected with possible
lack of control on the $L^p$ norm of the minimizers.
