  Let $\gamma > 0\,$, $\beta > 0\,$, $\alpha > 0$ and $0 < \sigma N < 4$. In
the present paper, we study, for $c > 0$ given, the constrained minimization
problem \begin{equation*} \label{MinL2fixed} m(c):=\inf_{u\in S (c) }E(u),
\end{equation*} where \begin{equation*} E
(u):=\frac{\gamma}{2}\int_{\mathbb{R}^N}|\Delta u|^2\, dx
-\frac{\beta}{2}\int_{\mathbb{R}^N}|\nabla u|^2\,
dx-\frac{\alpha}{2\sigma+2}\int_{\mathbb{R}^N}|u|^{2\sigma+2}\, dx,
\end{equation*} and \begin{equation*} S(c):=\left\{u\in
H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^{2}\, dx=c\right\}. \end{equation*}
The aim of our study is twofold. On one hand, this minimization problem is
related to the existence and orbital stability of standing waves for the mixed
dispersion nonlinear biharmonic Schr\"odinger equation \begin{equation*} i
\partial_t \psi -\gamma \Delta^2 \psi - \beta \Delta \psi + \alpha
|\psi|^{2\sigma} \psi =0, \quad \psi (0, x)=\psi_0 (x),\quad (t, x) \in
\mathbb{R} \times \mathbb{R}^N. \end{equation*} On the other hand, in most of
the applications of the Concentration-Compactness principle of P.-L. Lions, the
difficult part is to deal with the possible dichotomy of the minimizing
sequences. The problem under consideration provides an example for which, to
rule out the dichotomy is rather standard while, to rule out the vanishing,
here for $c > 0$ small, is challenging. We also provide, in the limit $c \to
0$, a precise description of the behaviour of the minima. Finally, some
extensions and open problems are proposed.
