  We study the mixed dispersion fourth order nonlinear Schr\"odinger equation
\begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma
\Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \R
\times\R^N, \end{equation*} where $\gamma,\sigma>0$ and $\beta \in \R$. We
focus on standing wave solutions, namely solutions of the form $\psi
(x,t)=e^{i\alpha t}u(x)$, for some $\alpha \in \R$. This ansatz yields the
fourth-order elliptic equation \begin{equation*}
%\tag{\protect{*}}\label{4nlsstar} \gamma \Delta^2 u -\beta \Delta u +\alpha u
=|u|^{2\sigma} u. \end{equation*} We consider two associated constrained
minimization problems: one with a constraint on the $L^2$-norm and the other on
the $L^{2\sigma +2}$-norm. Under suitable conditions, we establish existence of
minimizers and we investigate their qualitative properties, namely their sign,
symmetry and decay at infinity as well as their uniqueness, nondegeneracy and
orbital stability.
