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

Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schr\"odinger equation