  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).
