10 problems

Let $d\geq1$, $1+\frac{2}{d}<q<p<1+\frac{4}{d}$, and let $\rho_{0,E}>0$ be the threshold mass for the existence of ground states. Let $\rho^{\star}<\rho_{0,E}$ be the small-mass sc…

Strong instability conjecture. The standing wave $e^{i\lambda t}u^{-}(x)$ is strongly unstable.

Let $mu _{\beta}<0$ and assume the hypotheses of Theorem 2, under which $u^{-}$ is a normalized solution and $m_{\infty}$ is the ground-state energy of problem $(HE_{\infty})$. Sup…

Consider the one-dimensional nonlinear Schrödinger equation with double-power nonlinearity and its standing waves of frequency $\omega>0$, together with the corresponding algebraic…

Byeon–Huh–Seok conjecture. When $g>1$ and $m=0$, there are no static solutions.

Let $p\in(\frac{10}{3},6)$ and, for each $c>0$, let $\gamma(c)$ denote the mountain pass level associated with the prescribed-norm Schrödinger–Poisson problem. A value is a critica…

The discussion concerns branches of symmetric solutions, including the branch containing point B, the branch containing points D and E, and the main branch corresponding to symmetr…

Consider standing waves $u$ satisfying $u_{\pm\infty}=\pm1$ for the defocussing discrete nonlinear Schrödinger equation, with energy functional $\mathcal{E}$ restricted to the cons…

Equivalence conjecture. The optimization problems … have the same solution.

Let $e^{it\omega}\phi_{\omega}$ be a standing wave satisfying (H1-3), with $\phi_{\omega}$ real valued. Definition 2.3 gives the paper's new notion of linear stability. Orbital and…