Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent and a large frequency parameter in three and four dimensions

We prove the uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent and a large frequency parameter. This study is motivated by the paper [2] and aims to remove the restriction on dimension imposed there. In this paper, we employ the fixed-point argument developed in [7] to prove the uniquness. Hence, the linearization around the Aubin-Talenti function plays a key role. Furthermore, we need some estimates for the associated perturbed resolvents (see Proposition 3.1).