Solitons in the defocusing case

Let $\sigma=-1$. Consider the stationary nonlinear Schrödinger equation

$$Hvarphi+omegavarphi=-|\varphi|^2\varphi,$$

where $H$ has lowest eigenvalue $e_0<0$, and let $S$ denote the set of solitons, while $M$ is the mass, $E_0$ is the constrained ground-state energy, $E_1$ is the constrained excited-state energy, and $E_V$ is the energy associated with the potential $V$. Solitons in the defocusing case. The equation has a unique positive solution $\varphi_\omega$ for $omega i(0,-e_0)$. The soliton set is

$$S=\{e^{itheta}\varphi_\omega\band \omega i(0,-e_0),\theta iR\}.$$

The map $omega i(0,-e_0)mapstoM(varphi_omega)in(0,infty)$ is a monotone decreasing $C^1$ function. If $omega_0(mu)$ is its inverse, then $omega_0'(mu)<0$ and

$$E_0(mu)=E_V(\varphi_{\omega_0(mu)})in(e_0mu,0)\quad\text{for all }mu>0.$$

Moreover, $E_1(mu)=\infty$ for every $mu>0$. This conjecture describes the expected absence of excited solitons in the defocusing case and extends the corresponding radial result to the non-radial setting; the claimed existence, uniqueness, monotonicity, and energy characterization are not established here.