Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates

Abstract

We consider ground states solutions $u\in H^2({\mathbb{R}}^N)$ of biharmonic (fourth-order) nonlinear Schrödinger equations of the form $$ Δ^2 u + 2a Δu + b u - |u|^{p-2} u = 0 \quad \mbox{in $\mathbb{R}^N$} $$ with positive constants $a,b>0$ and exponents $2<p<2^{\ast }$, where $2^{\ast }=\frac{2N}{N-4}$ if $N>4$ and $2^{\ast }=\infty$ if $N\le 4$. By exploiting a connection to the adjoint Stein--Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states $u\in H^2({\mathbb{R}}^N)$ in dimension $N\ge 2$ fail to be radially symmetric for all exponents $2<p<\frac{2N+2}{N-1}$ in a suitable regime of $a,b>0$. As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained $L^2$-mass and for a related problem on the unit ball in ${\mathbb{R}}^N$ subject to Dirichlet boundary conditions.

arXiv commentSecond version with minor revisions, 17 pages