## Acta Mathematica

### Uniqueness of non-linear ground states for fractional Laplacians in ${\mathbb{R}}$

#### Abstract

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$where 0 < s < 1 and 0 < α < 4s/(1−2s) for ${s<\frac{1}{2}}$ and 0 < α <  for ${s\geq \frac{1}{2}}$. Here (−Δ)s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for ${s=\frac{1}{2}}$ and α = 1 in [5] for the Benjamin–Ono equation.

As a technical key result in this paper, we show that the associated linearized operator L+ = (−Δ)s+1−(α+1)Qα is non-degenerate; i.e., its kernel satisfies ker L+ = span{Q′}. This result about L+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.

#### Article information

Source
Acta Math., Volume 210, Number 2 (2013), 261-318.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892706

Digital Object Identifier
doi:10.1007/s11511-013-0095-9

Mathematical Reviews number (MathSciNet)
MR3070568

Zentralblatt MATH identifier
1307.35315

Rights

#### Citation

Frank, Rupert L.; Lenzmann, Enno. Uniqueness of non-linear ground states for fractional Laplacians in ${\mathbb{R}}$. Acta Math. 210 (2013), no. 2, 261--318. doi:10.1007/s11511-013-0095-9. https://projecteuclid.org/euclid.acta/1485892706

#### References

• Abdelouhab L., Bona J. L., Felland M., Saut J.-C.: Nonlocal models for nonlinear, dispersive waves. Phys. D, 40, 360–392 (1989)
• Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, 55. U.S. Government Printing Office, Washington, DC, 1972.
• Albert J.P., Bona J.L.: Total positivity and the stability of internal waves in stratified fluids of finite depth. IMA J. Appl. Math., 46, 1–19 (1991)
• Albert J.P., Bona J.L., Saut J.-C.: Model equations for waves in stratified fluids. Proc. Roy. Soc. London Ser. A, 453, 1233–1260 (1997)
• Amick C.J., Toland J.F.: Uniqueness and related analytic properties for the Benjamin–Ono equation—a nonlinear Neumann problem in the plane. Acta Math., 167, 107–126 (1991)
• Bañuelos R., Kulczycki T.: The Cauchy process and the Steklov problem. J. Funct. Anal., 211, 355–423 (2004)
• Bennett D.P., Brown R.W., Stansfield S.E., Stroughair J.D., Bona J.L.: The stability of internal solitary waves. Math. Proc. Cambridge Philos. Soc., 94, 351–379 (1983)
• Blumenthal R.M., Getoor R.K.: Some theorems on stable processes. Trans. Amer. Math. Soc., 95, 263–273 (1960)
• Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32, 1245–1260 (2007)
• Carmona R., Masters W.C., Simon B.: Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal., 91, 117–142 (1990)
• Chang S.-M., Gustafson S., Nakanishi K., Tsai T.-P.: Spectra of linearized operators for NLS solitary waves. SIAM J. Math. Anal., 39, 1070–1111 ((2007/08))
• Chang S.-Y.A., González M.d.M.: Fractional Laplacian in conformal geometry. Adv. Math., 226, 1410–1432 (2011)
• Chen W., Li C., Ou B.: Classification of solutions for an integral equation. Comm. Pure Appl. Math., 59, 330–343 (2006)
• Elgart A., Schlein B.: Mean field dynamics of boson stars. Comm. Pure Appl. Math., 60, 500–545 (2007)
• Frank, R. L., Lenzmann, E. & Seiringer, R., Uniqueness of radial solutions for the fractional Laplacian. Preprint, 2013. arXiv:1302.2652 [math.AP].
• Frank R. L., Seiringer R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal., 255, 3407–3430 (2008)
• Fröhlich J., Lenzmann E.: Blowup for nonlinear wave equations describing boson stars. Comm. Pure Appl. Math., 60, 1691–1705 (2007)
• González M.d.M.: Gamma convergence of an energy functional related to the fractional Laplacian. Calc. Var. Partial Differential Equations, 36, 173–210 (2009)
• Graham C.R., Zworski M.: Scattering matrix in conformal geometry. Invent. Math., 152, 89–118 (2003)
• Kenig C.E., Martel Y., Robbiano L.: Local well-posedness and blow-up in the energy space for a class of L2 critical dispersion generalized Benjamin–Ono equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, 853–887 (2011)
• Kwong M. K.: Uniqueness of positive solutions of Δuu + up = 0 in Rn. Arch. Ration. Mech. Anal., 105, 243–266 (1989)
• Li Y.A., Bona J.L.: Analyticity of solitary-wave solutions of model equations for long waves. SIAM J. Math. Anal., 27, 725–737 (1996)
• Li Y.Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. (JEMS), 6, 153–180 (2004)
• Lieb E.H., Yau H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys., 112, 147–174 (1987)
• Lin Z.: Instability of nonlinear dispersive solitary waves. J. Funct. Anal., 255, 1191–1224 (2008)
• Ma L., Zhao L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal., 195, 455–467 (2010)
• Maehara R.: The Jordan curve theorem via the Brouwer fixed point theorem. Amer. Math. Monthly, 91, 641–643 (1984)
• Majda A.J., McLaughlin D.W., Tabak E.G.: A one-dimensional model for dispersive wave turbulence. J. Nonlinear Sci., 7, 9–44 (1997)
• Maz′ya, V.G., Sobolev Spaces. Springer Series in Soviet Mathematics. Springer, Berlin– Heidelberg, 1985.
• McLeod K.: Uniqueness of positive radial solutions of Δu + f(u) = 0 in Rn. II. Trans. Amer. Math. Soc., 339, 495–505 (1993)
• McLeod K., Serrin J.: Uniqueness of positive radial solutions of Δu + f(u) = 0 in Rn. Arch. Ration. Mech. Anal., 99, 115–145 (1987)
• Reed, M. & Simon, B., Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York, 1978.
• Weinstein M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal., 16, 472–491 (1985)
• Weinstein M.I.: Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Comm. Partial Differential Equations, 12, 1133–1173 (1987)