Some non-homogeneous Gagliardo–Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation

We study the standing waves for a fourth-order Schrödinger 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 $\textit{mass-subcritical}$ and $\textit{mass-critical}$ cases. In the $\textit{mass super-critical}$ 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 $μ_0\in (0,+\infty)$, our results on "best" local minimizers are also optimal.