Linear non-degeneracy and uniqueness of the bubble solution for the critical fractional Hénon equation in $ \mathbb{R}^N $

Salomón Alarcón; Begoña Barrios; Alexander Quaas

doi:10.3934/dcds.2022178

Article Contents

2023, Volume 43, Issue 5: 1763-1786. Doi: 10.3934/dcds.2022178

This issue Previous Article New existence for the higher order prescribed curvature problem Next Article The Boussinesq systems on the background of a line solitary wave

Linear non-degeneracy and uniqueness of the bubble solution for the critical fractional Hénon equation in $ \mathbb{R}^N $

1.
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
2.
Departamento de Análisis Matemático, Universidad de La Laguna C/. Astrofísico Francisco Sánchez s/n, 38200 - La Laguna, Spain

^*Corresponding author: Salomón Alarcón
^*Corresponding author: Salomón Alarcón

Received: May 2022

Revised: October 2022

Early access: December 13, 2022

Published online: December 13, 2022

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We study the equation

$\begin{equation*} (-\Delta)^s u = |x|^{\alpha} u^{\frac{N+2s+2\alpha}{N-2s}}\mbox{ in }\mathbb{R}^N, \end{equation*} \quad\quad\quad\quad\quad\quad\text{(P)}$

where $ (-\Delta)^s $ is the fractional Laplacian operator with $ 0 < s < 1 $, $ \alpha>-2s $ and $ N>2s $. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (P) and, as a consequence, a uniqueness result of those solutions with Morse index equal to one. In particular, the ground state solution is unique. Our non-degeneracy result extends in the radial setting some known theorems done by Dávila, Del Pino and Sire (see [15, Theorem 1.1]), and Gladiali, Grossi and Neves (see [28, Theorem 1.3]).

Keywords:

Fractional partial differential equations,
uniqueness,
non-degeneracy,
critical Hénon equation.

Mathematics Subject Classification: Primary: 35R11; Secondary: 35A12, 35B07, 35B33.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	F. J. Almgren, Jr. and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773. doi: 10.1090/S0894-0347-1989-1002633-4.
[2]	A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of $\Delta u+u^{(N+2)/(N-2)} = 0$, the scalar curvature problem in $\mathbb{R}^N$, and related topics, J. Funct. Anal., 165 (1999), 117-149. doi: 10.1006/jfan.1999.3390.
[3]	T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom., 11 (1976), 573-598. doi: 10.4310/jdg/1214433725.
[4]	B. Barrios and A. Quaas, The sharp exponent in the study of the nonlocal Hénon equation in $\mathbb{R}^N$: A Liouville theorem and an existence result, Calc. Var. Partial Differential Equations, 59 (2020), 114. doi: 10.1007/s00526-020-01763-z.
[5]	W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc., 123 (1995), 1897-1905. doi: 10.2307/2161009.
[6]	C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[7]	L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.
[8]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[9]	E. A. Carlen and M. Loss, Extremals of functionals with competing symmetries, J. Funct. Anal., 88 (1990), 437-456. doi: 10.1016/0022-1236(90)90114-Z.
[10]	H. Chan, M. del Mar González, Y. Huang, E. Mainini and B. Volzone, Uniqueness of entire ground states for the fractional plasma problem, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 195, 42 pp, 2020, arXiv: 2003.01093. doi: 10.1007/s00526-020-01845-y.
[11]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[12]	E. A. Coddington and N. Levinson., The Poincaré-Bendixson theory of two-dimensional autonomous systems, Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955,389-403.
[13]	L. Damascelli and F. Pacella., Morse Index of Solutions of Nonlinear Elliptic Equations. Series: De Gruyter Series in Nonlinear Analysis and Applications, 30., Walter de Gruyter GmbH, Berlin/Boston, 2019. ISBN 978-3-11-053732-1.
[14]	J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.
[15]	J. Dávila, M. del Pino and Y. Sire, Non-degeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc., 141 (2013), 3865-3870. doi: 10.1090/S0002-9939-2013-12177-5.
[16]	A. DelaTorre and M. del Mar González, Isolated singularities for a semilinear equation for the fractional Laplacian arising in conformal geometry, Rev. Mat. Iberoam., 34 (2018), 1645-1678. doi: 10.4171/rmi/1038.
[17]	A. DelaTorre, M. del Pino, M. del Mar González and J. Wei, Delaunay-type singular solutions for the fractional Yamabe problem, Math. Ann., 369 (2017), 597-626. doi: 10.1007/s00208-016-1483-1.
[18]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[19]	P. L. Felmer and A. Quaas, On critical exponents for Pucci's extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 843-865. doi: 10.1016/S0294-1449(03)00011-8.
[20]	P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023.
[21]	P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746.
[22]	P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1350023. doi: 10.1142/S0219199713500235.
[23]	R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.
[24]	R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591.
[25]	R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99. doi: 10.1007/s00526-009-0302-x.
[26]	R. L. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, in: Spectral Theory, Function Spaces and Inequalities, Oper. Theory Adv. Appl. 219, B. M. Brown et al. (eds.), Birkhauser, Basel, 55-67, 2012. doi: 10.1007/978-3-0348-0263-5_4.
[27]	R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430. doi: 10.1016/j.jfa.2008.05.015.
[28]	F. Gladiali, M. Grossi and S. L. N. Neves, Nonradial solutions for the Hénon equation in $\mathbb{R}^N$, Adv. Math., 249 (2013), 1-36. doi: 10.1016/j.aim.2013.07.022.
[29]	G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606. doi: 10.1007/BF01171116.
[30]	G. H. Hardy and J. E. Littlewood, Notes on certain inequalities connected with the calculus of variations, J. London Math. Soc., 5 (1930), 34-39. doi: 10.1112/jlms/s1-5.1.34.
[31]	M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astron. Astrophys., 24 (1973), 229-238.
[32]	I. W. Herbst, Spectral theory of the operator $(p^2 + m^2)^{1/2} - Z e^{2}/r$, Comm. Math. Phys., 53 (1977), 285-294, https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-53/issue-3/Spectral-theory-of-the-operator-p2m212-Ze2r/cmp/1103900706.full.
[33]	Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180. doi: 10.4171/jems/6.
[34]	Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.
[35]	E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032.
[36]	R. Musina and A. I. Nazarov, Complete classification and nondegeneracy of minimizers for the fractional Hardy-Sobolev inequality, and applications, J. Differential Equations, 280 (2021), 292-314. doi: 10.1016/j.jde.2021.01.022.
[37]	M. Reed and B. Simon., Methods of Modern Mathematical Physics. IV. Analysis of Operators, , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.
[38]	O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.
[39]	S. L. Sobolev, On a theorem of functional analysis, Amer. Math. Soc. Transl. Ser. 2, 34 (1963), 39-68.
[40]	G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110 (1976), 353-372. doi: 10.1007/BF02418013.
[41]	R. Zhuo, W. Chen, X. Cui, and Z. Yuan, A Liouville theorem for the fractional Laplacian, preprint, 2014, arXiv: 1401.7402v1.