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In this paper, a nonclassical shell model is utilized to investigate the dynamic stability of functionally graded piezoelectric (FGP) nanoshells considering the thermo-electro-mechanical coupling effects. Also, the FGP nanoshells on elastic foundations are subjected to periodic axial loading. The theoretical formulations are derived by implementing the first-order shear deformation (FSD) shell theory into the nonlocal strain gradient theory (NSGT). Hamilton's principle is utilized to obtain the equations of motion and related boundary conditions of the present nanoshell model. Moreover, the Mathieu-Hill equations determining the instability regions are presented and analyzed based on Bolotin's method. Results demonstrate that the external electric voltage, the elastic foundations, the temperature rise, the static load factor, the power-law index and the small-scale parameters have significant influences on the dynamic stability responses of FGP nanoshells. [ABSTRACT FROM AUTHOR]