Abstract: By using the compactness method, we gaved the existence of solutions to the Cauchy problem of the energy-critical fractional  nonlinear Schrodinger equation and proved the existence of global solution to the Cauchy problem. By constructing the approximation equation and taking the limit of the solution sequence satisfying the approximation equation, the obtained limit function was the global weak solution of the energy-critical fractional nonlinear Schrodinger equation, and it was proved that the weak solution satisfied the energy inequality and mass conservation property.

Key words: nonlinear Schrodinger equation, energy-critical, fractional order, weak solution, compactness