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In this paper we consider the nonlinear fractional logarithmicSchr\"{o}dinger equation. By using a compactness method, we construct a uniqueglobal solution of the associated Cauchy problem in a suitable functionalframework. We also prove the existence of ground states as minimizers of theaction on the Nehari manifold. Finally, we prove that the set of minimizers isa stable set for the initial value problem, that is, a solution whose initialdata is near the set will remain near it for all time.

Existence and stability of standing waves for nonlinear fractional Schr\"odinger equation with logarithmic nonlinearity