Some notation in Orbital stability of standing waves for some nonlinear Schrödinger equations

Question

Picture below are from Orbital stability of standing waves for some nonlinear Schrödinger equations.

First, what is the means of the two red line ?

Besides, in the definition of $I_\lambda^\infty$, seemly, the $\lambda$ is unnecessary .

Answer 1

The notation $]a,b[$ is here meant to be the same thing as the open interval $(a,b)$. I believe this notation is meant to refer to the fact that $(a,b) = \{ (-\infty,a] \cup [b,\infty)\}^C$.

As for the $I_\lambda$ definition, it's hard to say for sure from just the two small pictures you've posted. For instance, it's not clear what $A$ and $E$ are and if / how they depends on any parameters. If I had to guess, though, I would say that there is a typo in the definition and the $\mu$ is supposed to be a $\lambda$, i.e. the infimum is taken over $u$ such that $|u|_{L^2}^2 = \lambda$.