Remarks on Nondegeneracy of Ground States for Quasilinear Schr\"odinger Equations

TL;DR

This paper confirms that all ground states of a specific quasilinear Schrödinger equation are nondegenerate for certain parameters, and explores properties of the associated linear operator.

Contribution

It provides an affirmative answer to the nondegeneracy of ground states for the equation, extending previous partial results and analyzing related linear operators.

Findings

All ground states are nondegenerate for 1<p<3.

Derived properties of the linear operator associated with ground states.

Extended understanding of the structure of solutions for the quasilinear Schrödinger equation.

Abstract

In this paper, we answer affirmatively the problem proposed by A. Selvitella in his paper "Nondegenracy of the ground state for quasilinear Schr\"odinger Equations" (see Calc. Var. Partial Differ. Equ., {\bf 53} (2015), pp 349-364): every ground state of equation \begin{eqnarray*}-\Delta u-u\Delta |u|^2+\omega u-|u|^{p-1}u=0&&\text{in }\mathbb{R}^N\end{eqnarray*} is nondegenerate for $1<p<3$, where $\omega>0$ is a given constant and $N\ge1$. We also derive further properties on the linear operator associated to ground states of above equation.