Abstract
We consider ground state solutions u ∈ H2(ℝN) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form
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(ℝN) 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 ℝN subject to Dirichlet boundary conditions.
Article PDF
Avoid common mistakes on your manuscript.
References
M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 87–92.
E. Berchio, F. Gazzola and T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math. 620 (2008), 165–183.
D. Bonheure, J.-B. Casteras, E. Moreira dos Santos and R. Nascimento, Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation, SIAM J. Math. Anal. 50 (2018), 5027–5071.
D. Bonheure, J.-B. Casteras, T. Gou and L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime, Trans. Amer. Math. Soc. 372 (2019), 2167–2212.
D. Bonheure, J.-B. Castéras, T. Gou and L. Jeanjean, Strong instability of ground states to a fourth order Schrödinger equation, Int. Math. Res. Not. IMRN 2019 (2019), 5299–5315.
T. Boulenger and E. Lenzmann, Blowup for biharmonic NLS, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), 503–544.
L. Bugiera, E. Lenzmann and J. Sok, On symmetry and uniqueness of ground states for linear and nonlinear elliptic PDEs, SIAM J. Math. Anal. 54 (2022), 6119–6135.
M. Christ and S. Shao, Existence of extremals for a Fourier restriction inequality, Anal. PDE 5 (2012), 261–312.
L. Fanelli, L. Vega and N. Visciglia, On the existence of maximizers for a family of restriction theorems, Bull. Lond. Math. Soc. 43 (2011), 811–817.
A. J. Fernández, L. Jeanjean, R. Mandel and M. Mariş, Non-homogeneous Gagliardo–Nirenberg inequalities in ℝNand application to a biharmonic non-linear Schrödinger equation, J. Differential Equations 330 (2022), 1–65.
A. Ferrero, F. Gazzola and T. Weth, Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl. (4) 186 (2007), 565–578.
G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math. 62 (2002), 1437–1462.
D. Foschi, Global maximizers for the sphere adjoint Fourier restriction inequality, J. Funct. Anal. 268 (2015), 690–702.
R. L. Frank, E. H. Lieb and J. Sabin, Maximizers for the Stein–Tomas inequality, Geom. Funct. Anal. 26 (2016), 1095–1134.
F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer, Berlin, 2010.
F. Gazzola and G. Sperone, Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations, Math. Eng. 4 (2022), 1–24.
V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D 144 (2000), 194–210.
E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, Int. Math. Res. Not. IMRN 2021 (2021), 15040–15081.
B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ. 4 (2007), 197–225.
B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal. 256 (2009), 2473–2517.
B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity 26 (2013), 2175–2191.
S. Shao, On existence of extremizers for the Tomas–Stein inequality for S1, J. Funct. Anal. 270 (2016), 3996–4038.
E. M. Stein, Oscillatory integrals in Fourier analysis, in Beijing Lectures in Harmonic Analysis (Beijing, 1984), Princeton University Press, Princeton, NJ, 1986, pp. 307–355.
G. Sweers. No Gidas–Ni–Nirenberg type result for semilinear biharmonic problems, Math. Nachr. 246/247 (2002), 202–206.
P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478.
T. H. Wolff. Lectures on Harmonic Analysis, American Mathematical Society, Providence, RI, 2003.
Acknowledgment
The second author would like to thank Antonio Fernández for valuable comments.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made (https://creativecommons.org/licenses/by/4.0/).
About this article
Cite this article
Lenzmann, E., Weth, T. Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates. JAMA 152, 777–800 (2024). https://doi.org/10.1007/s11854-023-0311-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-023-0311-2