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In this paper, we study a class of fractional Schr\"{o}dinger equationsinvolving logarithmic and critical nonlinearities on an unbounded domain, andshow that such an equation with positive or sign-changing weight potentialsadmits at least one positive ground state solution and the associated energy ispositive (or negative). By applying the Nehari manifold method andLjusternik-Schnirelmann category, we deeply investigate how the weightpotential affects the multiplicity of positive solutions, and obtain therelationship between the number of positive solutions and the category of somesets related to the weight potential.

Positive solutions for the fractional Schr\"{o}dinger equations with logarithmic and critical nonlinearities