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

Abstract

We consider ground state solutions u ∈ H²(ℝ^N) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form

$${\Delta ^2}u + 2a\Delta u + bu - |u{|^{p - 2}}u = 0\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N}$$

with positive constants a, b > 0 and exponents 2 < p < 2*, where \({2^ * } = {{2N} \over {N - 4}}\) if N > 4 and 2* = ∞ if N ≤ 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 ∈ H²(ℝ^N) in dimension N ≥ 2 fail to be radially symmetric for all exponents \(2 < p < {{2N + 2} \over {N - 1}}\) 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 L²-mass and for a related problem on the unit ball in ℝ^N subject to Dirichlet boundary conditions.