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

Abstract

Given $\rho >0$, we consider the problem \[ \text{find $(\lambda ,u)\in \mathbb{R}\times H_0^1(B)$ such that } \begin{cases} -Δu+λu = |u|^{p-1}u & \text{in } B \\ \int_B u^2\,dx = ρ, \end{cases} \] where $B$ is a ball in ${\mathbb{R}}^N$, $N\ge 1$, and $1<p<2^{\ast }-1$. 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.