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We consider axionlike particles, as the most probable constituents of darkmatter, the Yukawa-type corrections to Newton's gravitational law andconstraints on their parameters following from astrophysics and differentlaboratory experiments. After a brief discussion of the results by Prof. Yu. N.Gnedin in this field, we are coming to the recent experiment on measuring thedifferential Casimir force between Au-coated surfaces of a sphere and the topand bottom of rectangular trenches. In this experiment, the Casimir force wasmeasured over an unusually wide separation region from 0.2 to 8~mcm andcompared with the exact theory based on first principles of quantumelectrodynamics at nonzero temperature. We use the measure of agreement betweenexperiment and theory for obtaining constraints on the coupling constant ofaxionlike particles to nucleons and on the interaction strength of Yukawa-typeinteraction. The obtained constraints on the axion-to-nucleon coupling constantand on the strength of Yukawa interaction are stronger by the factors of 4 and24, respectively, than those found previously from gravitational experimentsand measurements of the Casimir force, but weaker than the constraintsfollowing from the differential measurement where the Casimir force wasnullified. Some other already performed and planned experiments directed to thesearch of axions and non-Newtonian gravity are discussed and their prospectsare evaluated.

Dark Matter Axions, Non-Newtonian Gravity and Constraints on them from Recent Measurement of the Casimir Force in the Micrometer Separation Range