Some remarks on a minimization problem associated to a fourth order nonlinear Schr\"odinger equation

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.