Normalized solutions to critical and supercritical Schrödinger equations in a ball

Abstract

In this paper, we study radial normalized solutions to nonlinear Schrödinger equations in the unit ball \[ -Δu+λu=|u|^{p-1}u, \qquad \int_{B_1}|u|^2\,dx=ρ, \] where $\lambda$ is unknown and $p\ge p_c:=\frac{N+2}{N-2}$. Soave \cite{Soave} proved that for every $1<p<p_c$ every positive mass is attained by infinitely many radial solutions and explicitly raised the question of whether every mass is attained in the Sobolev critical and supercritical regimes. We answer this question negatively in the radial class: for every $p\ge p_c$, the set of masses attained by nontrivial radial solutions is an interval with finite positive supremum. In particular, at $p=p_c$ the set of attainable masses is $(0,{\rho }_{\mathrm{rad}}^{\ast })$ or $(0,{\rho }_{\mathrm{rad}}^{\ast }]$, with $0<{\rho }_{\mathrm{rad}}^{\ast }<\infty$, and the same finite threshold alternative holds for every $p>p_c$. Thus, in the radial class, the subcritical all masses phenomenon fails throughout the full Sobolev critical and supercritical range.

arXiv comment19 pages