  In this article, we prove the existence of global weak solutions to the
three-dimensional focusing energy-critical nonlinear Schr\"odinger (NLS)
equation in the non-radial case. Furthermore, we prove the weak-strong
uniqueness for some class of initial data. The main ingredient of our new
approach is to use solutions of an energy-critical Ginzburg-Landau equation as
approximations for the corresponding nonlinear Sch\"ordinger equation.
  In our proofs, we first show the dichotomy of global well-posedness versus
finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1(
\mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the
stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle
[25,26], using a concentration-compactness/rigidity argument to reduce the
global well-posedness to the exclusion of a critical element. The critical
element is ruled out by dissipation of the Ginzburg-Landau equation, including
local smoothness, backwards uniqueness and unique continuation. The existence
of global weak solution of the three dimensional focusing energy-critical
nonlinear Schr\"odinger equation in the non-radial case then follows from the
global well-posedness of the energy-critical Ginzburg-Landau equation via a
limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove
the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is
bounded by a constant depending on the stationary solution $W$.
