  Before reaching their quiescent terminal white-dwarf cooling branch, some
low-mass helium-core white dwarf stellar models experience a number of nuclear
flashes which greatly reduce their hydrogen envelopes. Just before the
occurrence of each flash, stable hydrogen burning may be able to drive global
pulsations that could be relevant to shed some light on the internal structure
of these stars through asteroseismology. We present a pulsational stability
analysis applied to low-mass helium-core stars on their early white-dwarf
cooling branches going through CNO flashes in order to study the possibility
that the $\varepsilon$ mechanism is able to excite gravity-mode pulsations. We
carried out a nonadiabatic pulsation analysis for low-mass helium-core
white-dwarf models going through CNO flashes during their early cooling phases.
We found that the $\varepsilon$ mechanism due to stable hydrogen burning can
excite low-order ($\ell= 1, 2$) gravity modes with periods between $\sim 80$
and $500\ $s, for stars with $0.2025 \lesssim M_{\star}/M_{\odot} \lesssim
0.3630$ located in an extended region of the $\log g - T_{\rm eff}$ diagram
with effective temperature and surface gravity in the ranges $15\,000 \lesssim
T_{\rm eff} \lesssim 38\,000\ $K and $5.8 \lesssim \log g \lesssim 7.1$,
respectively. Since the timescales required for these modes to reach amplitudes
large enough to be observable are shorter than their corresponding evolutionary
timescales, the detection of pulsations in these stars is feasible. If a
low-mass white dwarf star were found to pulsate with low-order gravity modes in
this region of instability, it would confirm our result that such pulsations
can be driven by the $\varepsilon$ mechanism. In addition, confirming a rapid
rate of period change in these pulsations would support that these stars
actually experience CNO flashes, as predicted by evolutionary calculations.
