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

TL;DR

This paper proves that ground state solutions of a biharmonic nonlinear Schrödinger equation in dimensions two and higher are not radially symmetric for certain exponents, using Fourier extension estimates and symmetry breaking techniques.

Contribution

It introduces a novel symmetry breaking result for ground states of biharmonic NLS equations by connecting to Stein--Tomas inequalities and trial functions, extending understanding of symmetry properties.

Findings

Ground states fail to be radially symmetric for specific exponents.

Symmetry breaking occurs in a broad regime of parameters.

Results apply to constrained minimization and boundary value problems.

Abstract

We consider ground states solutions $u \in H^2(\mathbb{R}^N)$ of biharmonic (fourth-order) nonlinear Schr\"odinger equations of the form $$ \Delta^2 u + 2a \Delta u + b u - |u|^{p-2} u = 0 \quad \mbox{in $\mathbb{R}^N$} $$ with positive constants $a, b > 0$ and exponents $2 < p < 2^*$, where $2^* = \frac{2N}{N-4}$ if $N > 4$ and $2^* = \infty$ if $N \leq 4$. 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 $N \geq 2$ 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.