4 problems

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 Schrö…

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…

For $d=2$, let $Q_\omega$ denote a cubic-quintic ground state solution at frequency $\omega\in(0,\tfrac{3}{16})$. Orbital stability conjecture. All cubic-quintic ground state solut…

In dimension $d=3$, let $\phi_\omega$ denote the ground state solution of the cubic-quintic Schrödinger equation at frequency $\omega$, with $0<\omega<\tfrac{3}{16}$, and let…