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

• Published in: Journal d'analyse mathématique (Jerusalem) 2024, Vol.152 (2), p.777-800
• Main Authors: Lenzmann, Enno, Weth, Tobias
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Analysis; Boundary conditions; Broken symmetry; Dirichlet problem; Dynamical Systems and Ergodic Theory; Exponents; Functional Analysis; Ground state; Mathematics; Mathematics and Statistics; Partial Differential Equations; Schrodinger equation; Symmetry
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-023-0311-2

We consider ground state solutions u ∈ H 2 (ℝ N ) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form 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 ∗ = 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 ∈ H 2 (ℝ N ) in dimension N ≥ 2 fail to be radially symmetric for all exponents 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 L 2 -mass and for a related problem on the unit ball in ℝ N subject to Dirichlet boundary conditions.