Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

TL;DR

This paper extends the analysis of ground state solutions to high-frequency regimes in high-dimensional semilinear elliptic equations involving Sobolev critical exponents, revealing their behavior and degeneracy properties.

Contribution

It generalizes previous results to high frequencies for dimensions five and higher, characterizing the solutions' behavior and degeneracy in these regimes.

Findings

Ground states are explicit and well-characterized at high frequencies.

Degeneracy of solutions is analyzed and understood in high-dimensional settings.

Main behavior is dominated by Sobolev critical nonlinearities.

Abstract

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [AIKN3], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schr\"odinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.