On the Number of Normalized Ground State Solutions for a class of Elliptic Equations with general nonlinearities and potentials

TL;DR

This paper characterizes the set of normalized ground state solutions for a class of elliptic equations with general nonlinearities and potentials, establishing conditions for uniqueness and smooth dependence on mass.

Contribution

It provides a detailed description of the uniqueness and structure of normalized ground state solutions for elliptic equations with general nonlinearities and potentials.

Findings

NGSS is unique for all but finitely many masses under certain conditions.

When unique, NGSS varies smoothly with the mass.

The method applies broadly to similar equations with key properties.

Abstract

We provide a precise description of the set of normalized ground state solutions (NGSS) for the class of elliptic equations: $$ -\Delta u - \lambda u + V (| x |) u - f (| x |, u) = 0,\quad\text{in}\quad \mathbb{R}^n,\ n\geq 1. $$ In particular, we show that under suitable assumptions on $V$ and $f$, the NGSS is unique for all the masses except for at most a finite number. Moreover, we prove that when unique, the NGSS $u_c$ is a smooth function of the mass $c.$ Our method is as follow: using the NGSS for a given mass $c$, we construct an exhaustive list of potential candidates to the minimization problem for masses close to $c$, and we develop a strategy how to pick the right one. In particular, if there is a unique NGSS for a given mass $c_0,$ then this uniqueness property is inherited for all the masses $c$ close to $c_0.$ Our method is general and applies to other equations provided that some key properties hold true.