In this paper, we consider blow-up behaviors of constraint minimizers for the \(\dot{H}^{\gamma _c}\) -critical fourth-order nonlinear Schrödinger equation with the mixed dispersions \(\begin{aligned} i\psi _t -\Delta ^2 \psi +\mu \Delta \psi + |\psi |^{p}\psi =0. \end{aligned}\) This equation arises in describing the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. This paper seems to be the first time to present and study the minimizing problem: \(\begin{aligned} m_1(c):=\inf \left\{ E(u),~~u\in \dot{H}^{\gamma _c}\cap \dot{H}^2~and~\Vert u\Vert _{\dot{H}^{\gamma _c}}=c\right\} , \end{aligned}\) where \(\gamma _c:=\frac{N}{2}-\frac{4}{p}\) is the critical Sobolev exponent and \(\begin{aligned} E(u)=\frac{1}{2}\Vert \Delta u\Vert _{L^2}^2+\frac{\mu }{2}\Vert \nabla u\Vert _{L^2}^2-\frac{1}{p+2}\Vert u\Vert _{L^{p+2}}^{p+2}. \end{aligned}\) Minimizers of this problem exist only if \(c<\Vert Q\Vert _{\dot{H}^{\gamma _c}}\) , where Q is a solution of equation \(\Delta ^2Q +(-\Delta )^{\gamma _c}Q-|Q|^{p}Q=0\) . We then give a detailed description of blow-up behavior of minimizers as \(c\nearrow \Vert Q\Vert _{\dot{H}^{\gamma _c}}\) .
