  The stability of a quantum critical point in the $O(N)$ universality class
with respect to an elastic coupling, that preserves $O(N)$ symmetry, is
investigated for isotropic elasticity in the framework of the renormalization
group (RG) close to the upper critical dimension $d=3-\epsilon$. With respect
to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant
in the RG sense for $1\leq N \leq 4$, and the crystal becomes microscopically
unstable, i.e., a sound velocity vanishes at a finite value of the correlation
length $\xi$. For $N > 4$, an additional fixed point emerges that is located at
a finite value of the dimensionless elastic coupling. This fixed point is
repulsive and separates the flow to weak and strong elastic coupling. As the
fixed point is approached the sound velocity is found to vanish only
asymptotically as $\xi \to \infty$ such that the crystal remains
microscopically stable for any finite value of $\xi$. The fixed point structure
we find for the quantum problem is distinct from the classical counterpart in
$d=4-\epsilon$, where the crystal always remains microscopically stable for
finite $\xi$.
