7 problems

Consider the nonlinear Schrödinger equation in the setting of the paper, and let $Q$ denote the ground state with critical mass $\|Q\|_{L^2}^2$. Let the symmetries of the equation…

Stability conjecture. Contracting sphere blow-up solutions are stable under small radial perturbations but unstable under small nonradial (symmetry-breaking) perturbations.

Low-index nonexistence conjecture. There are no solutions of $\mathrm{HSE}$ satisfying $\mathrm{HBC1}$--$\mathrm{HBC2}$ when $\nu(n,\sigma)\leq2$.

Slow blow-up existence conjecture. The analysis of He and Velázquez can be used to prove the existence of slow blow-up solutions for $n>4$.

Consider general blow-up solutions of the nonlinear Schrödinger equation, without imposing the special pseudo-conformal multi-bubble form. Merle–Raphaël multi-bubble decomposition…

Consider a blow-up solution of the nonlinear Schrödinger equation with singularities, and let $m_k$ be the mass concentrated at the $k$-th singularity, with $m_k\geq \|Q\|_{L^2}^2$…

Sire–Wei–Zheng's conjecture. Bubble tower solutions to with $b=0$ should exist.