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

Abstract

We provide a precise description of the set of normalized ground state solutions (NGSS) for the class of elliptic equations: $$-\Delta u-\lambda u+V(|x|)u-f(|x|,u)=0,\text{in}{\mathbb{R}}^n,n\ge 1.$$ 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.