Numerical slope conjecture for ground states of the nonlinear Schrödinger equation on the unit ball

Let $B_1$ be the unit ball, let $\beta$ denote the first Dirichlet eigenvalue parameter, and let $Q_b$ be the least-energy (ground state) solution of the stationary nonlinear Schrödinger equation for $b\in(-\lambda_1,\infty)$. Write $M(Q_b)$ for its mass, let $\mathcal R$ be the corresponding ground state on the whole space, and let $u(t)$ denote the solution with initial condition $u_0$. The standing wave associated with $Q_b$ is $e^{ibt}Q_b$. Slope conjecture. For $d\geq1$: (I) if $0<\alpha<4/d$, then for every $b\in(-\lambda_1,\infty)$,

$$\partial_bM(Q_b)>0,$$

and $e^{ibt}Q_b$ is orbitally stable in $H_0^1(B_1)$; (II) if $\alpha=4/d$, the same positivity and orbital stability hold for every $b\in(-\lambda_1,\infty)$, $M(Q_b)$ increases monotonically with

$$\lim_{b\to\infty}M(Q_b)=M(\mathcal R),$$

and initial data satisfying $M(u_0)>M(\mathcal R)$ produce solutions that blow up in finite time; (III) if $\alpha>4/d$ (with $\alpha<4/(d-2)$ when $d\geq3$), there is $b^\ast\in(-\lambda_1,\infty)$ such that $M(Q_b)$ increases to a maximum at $b^\ast$ and then decreases to $M(\mathcal R)$, with $\partial_bM(Q_b)>0$ and orbital stability for $b\in(-\lambda_1,b^\ast)$, and $\partial_bM(Q_b)<0$ and instability for $b\in(b^\ast,\infty)$. The unstable branch may lead either to finite-time blow-up or to oscillation between two states. This conjecture is motivated by the Vakhitov–Kolokolov slope condition and the Grillakis–Shatah–Strauss stability method. The source says that these claims are confirmed numerically; the parser supplies no independent resolution status, so the conjecture is recorded as open.