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We consider nonlinear scalar field equations involving the Sobolev-criticalexponent at high frequencies $\omega$. Since the limiting profile of the groundstate as $\omega \to \infty$ is the Aubin-Talenti function and degenerate in acertain sense, from the point of view of perturbation methods, thenondegeneracy problem for the ground states at high frequencies is subtle. Inaddition, since the limiting profile (Aubin-Talenti function) fails to lie in$L^{2}(\mathbb{R}^{d})$ for $d=3,4$, the nondegeneracy problem for $d=3,4$ ismore difficult than that for $d\ge 5$ and an applicable methodology is notknown. In this paper, we solve the nondegeneracy problem for $d=3,4$ bymodifying the arguments in [2, 3]. We also show that the linearized operatoraround the ground state has exactly one negative eigenvalue.

Nondegeneracy of ground states for nonlinear scalar field equations  involving the Sobolev-critical exponent at high frequencies in three and four  dimensions