4 problems

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…

Let $\blambda_c$ be the Lagrange multiplier associated with the mountain pass solution at mass $c$. Mountain pass multiplier conjecture. One has … The paper proves that the multipl…

Let a standing wave be a standing-wave solution associated with a critical point of the constrained functional, and distinguish mountain pass solutions from local minimizers. Orbit…

Uniqueness conjecture. Then $lambda_1=lambda_2$ and $r_2=r_1(\cdot+\xi)$ for some $xi\in\mathbb{R}^3$.