Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schr\"odinger equations

TL;DR

This paper proves the non-degeneracy of certain non-radial sign-changing solutions to a nonlinear Schrödinger equation, marking the first such example with finite energy, which advances understanding of solution stability.

Contribution

It establishes the non-degeneracy of specific non-radial sign-changing solutions, providing the first finite energy example in this context.

Findings

Non-radial sign-changing solutions are non-degenerate.

First example of finite energy non-degenerate sign-changing solutions.

Results contribute to stability analysis of nonlinear Schrödinger solutions.

Abstract

We prove that the non-radial sign-changing solutions to the nonlinear Schr\"odinger equation \begin{equation*} \Delta u-u+|u|^{p-1}u=0 \mbox{ in }\R^N, \quad u \in H^1 (\R^N ) \end{equation*} constructed by Musso, Pacard and Wei is non-degenerate. This provides the first example of non-degenerate sign-changing solution with finite energy to the above nonlinear Schr\"odinger equation.