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

Abstract

AbstractWe consider ground state solutions u ∈ H2(ℝN) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form $${\Delta ^2}u + 2a\Delta u + bu - |u{|^{p - 2}}u = 0\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N}$$ Δ 2 u + 2 a Δ u + b u − ∣ u ∣ p − 2 u = 0 in ℝ N with positive constants a, b > 0 and exponents 2 < p < 2*, where $${2^ * } = {{2N} \over {N - 4}}$$ 2 ∗ = 2 N 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(ℝN) in dimension N ≥ 2 fail to be radially symmetric for all exponents $$2 < p < {{2N + 2} \over {N - 1}}$$ 2 < p < 2 N + 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 L2-mass and for a related problem on the unit ball in ℝN subject to Dirichlet boundary conditions.