Stability and ground-state transition conjecture for solitary waves

Let $d$ be the spatial dimension, let $\alpha$ be the nonlinearity exponent, and let $\varphi_\omega$ be the solitary-wave profile associated with frequency $\omega\in(0,\omega^*)$. Write $M(\omega)$ for its mass and $\mathcal{E}$ for the energy. A solitary wave is orbitally stable when its orbit under the symmetries of the evolution remains close to the initial orbit for all later times.

Stability conjecture. Let $1\leq d\leq 6$. For $\alpha\leq 2/d$, the solution $\varphi_\omega$ minimizes $\mathcal{E}$ at mass $M=M(\omega)$ for all $\omega\in(0,\omega^*)$ and is orbitally stable. For $\alpha>2/d$, there exists an $\omega_\mathrm{c}\in(0,\omega^*)$ such that, for $\omega\geq\omega_\mathrm{c}$, $\varphi_\omega$ is a normalized ground state and orbitally stable, whereas for $\omega<\omega_\mathrm{c}$ it is not a minimizer of $\mathcal{E}$ at fixed mass $M=M(\omega)$ and is unstable.

This conjecture predicts the stability regions suggested by the monotonicity of the mass-frequency function and the numerical experiments. It remains open in the stated generality, since the paper presents the conclusion as an expectation rather than an established theorem.