Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter

TL;DR

This paper proves that for large frequency parameters, the ground state of combined power-type nonlinear scalar field equations with Sobolev critical exponent is nondegenerate and linearly unstable, with a negative derivative of mass w.r.t. frequency.

Contribution

It establishes nondegeneracy and linear instability of the ground state for large frequencies, extending previous results for small frequencies.

Findings

Ground state is nondegenerate for large frequency.

Ground state exhibits linear instability at large frequency.

Derivative of mass with respect to frequency is negative.

Abstract

We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In \cite{AIKN3}, it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schr\"odinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.