  We consider nonlinear scalar field equations involving the Sobolev-critical
exponent at high frequencies $\omega$. Since the limiting profile of the ground
state as $\omega \to \infty$ is the Aubin-Talenti function and degenerate in a
certain sense, from the point of view of perturbation methods, the
nondegeneracy problem for the ground states at high frequencies is subtle. In
addition, since the limiting profile (Aubin-Talenti function) fails to lie in
$L^{2}(\mathbb{R}^{d})$ for $d=3,4$, the nondegeneracy problem for $d=3,4$ is
more difficult than that for $d\ge 5$ and an applicable methodology is not
known. In this paper, we solve the nondegeneracy problem for $d=3,4$ by
modifying the arguments in [2, 3]. We also show that the linearized operator
around the ground state has exactly one negative eigenvalue.
