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

Published online by Cambridge University Press:  07 May 2019

Takafumi Akahori ,
Slim Ibrahim  and
Hiroaki Kikuchi
Takafumi Akahori
Affiliation:
Faculty of Engineering, Shizuoka University, Jyohoku 3-5-1, Hamamatsu, Shizuoka432-8561, Japan (akahori.takafumi@shizuoka.ac.jp)
Slim Ibrahim
Affiliation:
Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, B.C., Canada (ibrahims@uvic.ca)
Hiroaki Kikuchi
Affiliation:
Department of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo187-8577, Japan (hiroaki@tsuda.ac.jp)
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Abstract

We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In [3], 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 Schrödinger 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.

Keywords

Linear instabilitynondegeneracyground statenonlinear scalar field equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2417 - 2441
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Copyright
Copyright © Royal Society of Edinburgh 2019

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