Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation

Abstract: In this paper, we consider the nonlinear Schr\"odinger equation $iu_t +\Delta u= \lambda |u|^{\frac {4} {N-4}} u$ in $\R^N $, $N\ge 5$, with $\lambda \in \C$. We prove local well-posedness (local existence, unconditional uniqueness, continuous dependence) in the critical space $\dot H^2 (\R^N) $.