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

TL;DR

This paper proves the nondegeneracy of ground states for nonlinear scalar field equations with Sobolev-critical exponent at high frequencies in three and four dimensions, addressing challenges due to the limiting profile's properties.

Contribution

It establishes the nondegeneracy of ground states in low dimensions by adapting existing methods, and analyzes the spectral properties of the linearized operator.

Findings

Nondegeneracy of ground states for d=3,4 established

Linearized operator has exactly one negative eigenvalue

Methodology adapted from previous works to handle Sobolev-critical cases

Abstract

We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies $\omega$. Since the limiting profile of the ground state as $\omega \to \infty$ 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 $L^{2}(\mathbb{R}^{d})$ for $d=3,4$, the nondegeneracy problem for $d=3,4$ is more difficult than that for $d\ge 5$ and an applicable methodology is not known. In this paper, we solve the nondegeneracy problem for $d=3,4$ by modifying the arguments in [2, 3]. We also show that the linearized operator around the ground state has exactly one negative eigenvalue.