Blow up dynamics for the 3D energy-critical nonlinear Schrödinger equation

Abstract We construct a two-parameter continuum of type II blow up solutions for the energy-critical nonlinear Schrödinger equation in dimension $$ d = 3$$ d = 3 . The solutions collapse to a single energy bubble in finite time and have the form $$\begin{aligned} u(t,x) = e^{i \alpha (t)}\lambda (t)^{\frac{1}{2}}W(\lambda (t) x) + \eta (t, x ),~ ~ t \in [0, T),~ x \in {{\,\mathrm{\mathbb {R}}\,}}^3, \end{aligned}$$ u ( t , x ) = e i α ( t ) λ ( t ) 1 2 W ( λ ( t ) x ) + η ( t , x ) , t ∈ [ 0 , T ) , x ∈ R 3 , where $$ W( x) = \big ( 1 + \frac{|x|^2}{3}\big )^{-\frac{1}{2}}$$ W ( x ) = ( 1 + | x | 2 3 ) - 1 2 is the ground state solution, $$\lambda (t) = (T -t)^{- \frac{1}{2} - \nu } $$ λ ( t ) = ( T - t ) - 1 2 - ν for suitable