The　structural　instability　of　multi-walled　carbon　nanotubes　(MWCNTs)　has　captured　extensive　attention　due　to　the　unique　characteristic　of　extremely　thin　hollow　cylinder　structure.　The　previous　studies　usually　focus　on　the　buckling　behavior　without　considering　the　effects　of　the　wall　number　and　initial　pressure.　In　this　paper,　the　axial　buckling　behavior　of　MWCNTs　with　the　length-to-outermost　radius　ratio　less　than　20　is　investigated　within　the　framework　of　the　Donnell　shell　theory.　The　governing　equations　for　the　infinitesimal　buckling　of　MWCNTs　are　established,　accounting　for　the　van　der　Waals　(vdW)　interaction　between　layers.　The　effects　of　the　wall　number,　initial　pressure　prior　to　buckling,　and　aspect　ratio　on　the　critical　buckling　mode,　buckling　load,　and　buckling　strain　are　discussed,　respectively.　Specially,　the　four-walled　and　twenty-walled　CNTs　are　studied　in　detail,　indicating　the　fact　that　the　buckling　instability　may　occur　in　other　layers　besides　the　outermost　layer.　The　obtained　results　extend　the　buckling　analysis　of　the　continuum-based　model,　and　provide　theoretical　support　for　the　application　of　CNTs.