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

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