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

TL;DR

This paper proves the uniqueness of ground states for certain nonlinear scalar field equations with critical exponents and large frequency parameters in three and four dimensions, extending previous results by removing dimensional restrictions.

Contribution

It introduces a fixed-point approach combined with linearization techniques around Aubin-Talenti functions to establish uniqueness in higher dimensions.

Findings

Proves ground state uniqueness in 3D and 4D.

Employs fixed-point and linearization methods.

Provides resolvent estimates for perturbed operators.

Abstract

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).