The double-power nonlinear Schr\"odinger equation and its generalizations: uniqueness, non-degeneracy and applications

TL;DR

This paper proves the uniqueness and non-degeneracy of positive radial solutions to a class of nonlinear Schrödinger equations with double power non-linearity, analyzes their behavior in certain limits, and discusses implications for energy minimizers and stability.

Contribution

It establishes a general uniqueness and non-degeneracy result for solutions of nonlinear Schrödinger equations and applies it specifically to the double power case, including behavior analysis and conjectures.

Findings

Uniqueness of solutions in the double power case.

Non-degeneracy of the solutions allows asymptotic analysis.

Numerical simulations support the conjecture on mass variation.

Abstract

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $\Delta u+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_\mu$ allows us to derive its behavior in the two limits $\mu\to0$ and $\mu\to\mu_*$ where $\mu_*$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the $L^2$ mass of $u_\mu$ in terms of $\mu$, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of $u_\mu$.