  We study the standing waves for a fourth-order Schr\"odinger equation with
mixed dispersion that minimize the associated energy when the $L^2-$norm (the
\textit{mass}) } is kept fixed. We need some non-homogeneous
Gagliardo-Nirenberg-type inequalities and we develop a method to prove such
estimates that should be useful elsewhere. We prove optimal results on the
existence of minimizers in the {\it mass-subcritical } and {\it mass-critical }
cases. In the { \it mass supercritical} case we show that global minimizers do
not exist, and we investigate the existence of local minimizers. If the mass
does not exceed some threshold $ \mu_0 \in (0,+\infty)$, our results on "best"
local minimizers are also optimal.
