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

We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies ω. Since the limiting profile of the ground states as ω→∞ 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 L2(Rd) for d=3,4, the nondegeneracy problem for d=3,4 is more difficult than that for d≥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 Akahori et al. (2020) and Akahori et al. (2019). We also show that the linearized operator around the ground state has exactly one negative eigenvalue.