2021/10/20 by Enno Lenzmann, Lenzmann, Enno, Tobias Weth +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2110.10782
openalex publication_date 2021/10/20 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We consider ground states solutions u ∈ H2(ℝN) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form Δ2 u + 2a Δu + b u - |u|p-2 u = 0 in ℝN with positive constants a, b > 0 and exponents 2 < p < 2^*, where 2^* = (2N)/(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)/(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.