Normalized solutions to the NLS equation in the ball for any prescribed mass

Abstract

Given $\rho>0$, we consider the problem \[ \text{find $(\lambda,u) \in \mathbb{R} \times H_0^1(B)$ such that } \begin{cases} -\Delta u+\lambda u = |u|^{p-1}u&\text{in } B \\ \int_B u^2\,dx = \rho, \end{cases} \] where $B$ is a ball in $\mathbb{R}^N$, $N \ge 1$, and $1<p<2^*-1$. Without any further restriction on $N$, $\rho$ and $p$, we prove the existence of infinitely many radial solutions, and, in dimension $N \ge 4$, of at least one non-radial solution.