  In this paper, we study a class of fractional Schr\"{o}dinger equations
involving logarithmic and critical nonlinearities on an unbounded domain, and
show that such an equation with positive or sign-changing weight potentials
admits at least one positive ground state solution and the associated energy is
positive (or negative). By applying the Nehari manifold method and
Ljusternik-Schnirelmann category, we deeply investigate how the weight
potential affects the multiplicity of positive solutions, and obtain the
relationship between the number of positive solutions and the category of some
sets related to the weight potential.
