The dynamic stability of single-walled carbon nanotubes (SWCNT) and double-walled carbon nanotubes (DWCNT) embedded in an elastic medium subjected to combined static and periodic axial loads are investigated using Floquet–Lyapunov theory and bounded solution theory. An elastic Euler- Bernoulli beam model is utilized in which the nested slender nanotubes are coupled with each other through the van der Waals (vdW) interlayer interaction. The Galerkin’s approximate method on the basis of trigonometric mode shape functions is applied to reduce the coupled governing partial differential equations to a system of the extended Mathieu-Hill equations. Applying Floquet–Lyapunov theory and Rung-Kutta numerical integration method with Gill coefficients, the influences of number of layer, elastic medium, exciting frequency and combination of exciting frequency on the instability conditions of SWCNTs and DWCNTs are investigated. A satisfactory agreement can be observed by comparison between the predicted results of Floquet–Lyapunov theory with bounded solutions theory ones. Based on results, increasing the number of layers, and elastic medium, dynamic stability of SWCNTs and DWCNTs surrounding elastic medium increase. Moreover, the instability of CNTs increases by increasing the exciting frequency.