Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Liebs Translation Lemma and a Riesz energy version of the Brezis--Lieb lemma.