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

Abstract

We consider ground state solutions u ∈ H2(ℝN) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form Δ2u+2aΔu+bu−∣u∣p−2u=0inℝN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Delta ^2}u + 2a\\Delta u + bu - |u{|^{p - 2}}u = 0\\,\\,\\,\\,{\\rm{in}}\\,\\,{\\mathbb{R}^N}$$\\end{document} with positive constants a, b > 0 and exponents 2 < p < 2*, where 2∗=2NN−4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${2^ * } = {{2N} \\over {N - 4}}$$\\end{document} 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+2N−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 < p < {{2N + 2} \\over {N - 1}}$$\\end{document} 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.