Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation

Previous Article
Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion
DCDS Home
This Issue
Next Article
Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

June 2020, 40(6): 3629-3656. doi: 10.3934/dcds.2020032

Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation

Joel Kübler and Tobias Weth ^,

Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, D-60629 Frankfurt a.M., Germany

^* Corresponding author: Tobias Weth

Dedicated to Wei-Ming Ni with admiration and appreciation

Received January 2019 Revised June 2019 Published October 2019

Abstract
Full Text(HTML)
Related Papers
Under a Creative Commons license

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem

$ \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -\Delta u & = |x|^\alpha |u|^{p-2}u&&\qquad \text{in $ {\bf B}$,}\\ u& = 0&&\qquad \text{on $\partial {\bf B}$,} \end{aligned} \right. \end{equation*} $

in the unit ball

$ {\bf B} \subset \mathbb{R}^N,N\geq 3 $

,

$ p>2 $

in the limit

$ \alpha \to +\infty $

. More precisely, for a given positive integer

$ K $

, we derive asymptotic

$ C^1 $

-expansions for the negative eigenvalues of the linearization of the unique radial solution

$ u_\alpha $

of

$ (H) $

with precisely

$ K $

nodal domains and

$ u_\alpha(0)>0 $

. As an application, we derive the existence of an unbounded sequence of bifurcation points on the radial solution branch

$ \alpha \mapsto (\alpha,u_\alpha) $

which all give rise to bifurcation of nonradial solutions whose nodal sets remain homeomorphic to a disjoint union of concentric spheres.

Keywords: Hénon equation, Morse index, spectral asymptotics, bifurcation, nodal solutions.

Mathematics Subject Classification: 35J25, 35B32, 35P15.

Citation: Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

References:

[1]	A. Amadori and F. Gladiali, Bifurcation and symmetry breaking for the Hénon equation, Adv. Differential Equations, 19 (2014), 755-782.
[2]	A. Amadori and F. Gladiali, On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's, preprint, arXiv: 1805.04321
[3]	A. Amadori and F. Gladiali, Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem, preprint, arXiv: 1810.11046
[4]	J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states, Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828. doi: 10.1016/j.anihpc.2006.04.001.
[5]	J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states, Ⅱ, J. Differential Equations, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.
[6]	D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.
[7]	E. N. Dancer, F. Gladiali and M. Grossi, On the Hardy-Sobolev equation, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 299-336. doi: 10.1017/S0308210516000135.
[8]	E. N. Dancer and J. C. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscripta Math., 123 (2007), 493-511. doi: 10.1007/s00229-007-0110-6.
[9]	F. Gladiali, M. Grossi and S. L. 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.
[10]	F. Gladiali, M. Grossi, F. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differential Equations, 40 (2011), 295-317. doi: 10.1007/s00526-010-0341-3.
[11]	M. Hénon, Numerical Experiments on The Stability of Spherical Stellar Systems, Astronomy and Astrophysics, 1973.
[12]	H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Mathematical Sciences, Springer-Verlag, New York, 2004. doi: 10.1007/b97365.
[13]	H. Kielhöfer, A bifurcation theorem for potential operators, J. Funct. Anal., 77 (1988), 1-8. doi: 10.1016/0022-1236(88)90073-0.
[14]	Z. Lou, T. Weth and Z. Zhang, Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type, Z. Angew. Math. Phys., 70 (2019), Art. 35, 19 pp, arXiv: 1803.02712. doi: 10.1007/s00033-019-1080-8.
[15]	E. Moreira dos Santos and F. Pacella, Morse index of radial nodal solutions of Hénon type equations in dimension two, Commun. Contemp. Math., 19 (2017), 1650042, 16 pp. doi: 10.1142/S0219199716500425.
[16]	K. Nagasaki, Radial solutions for $\Delta u + \|x\|^l \|u\|^{p-1}u = 0$ on the unit ball in $\mathbb{R}^n$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 211-232.
[17]	Y. Naito, Bounded solutions with prescribed numbers of zeros for the Emden-Fowler differential equation, Hiroshima Math. J., 24 (1994), 177-220. doi: 10.32917/hmj/1206128140.
[18]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.
[19]	A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97. doi: 10.1007/s00209-006-0060-9.
[20]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326. doi: 10.1007/s00526-004-0302-9.
[21]	D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.
[22]	D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[23]	E. Yanagida, Structure of radial solutions to $\Delta u + K(\|x\|)\|u\|^{p-1}u=0$ in $\mathbb{R}^N$, SIAM J. Math. Anal., 27 (1996), 997-1014. doi: 10.1137/0527053.

show all references

References:

[1]	A. Amadori and F. Gladiali, Bifurcation and symmetry breaking for the Hénon equation, Adv. Differential Equations, 19 (2014), 755-782.
[2]	A. Amadori and F. Gladiali, On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's, preprint, arXiv: 1805.04321
[3]	A. Amadori and F. Gladiali, Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem, preprint, arXiv: 1810.11046
[4]	J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states, Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828. doi: 10.1016/j.anihpc.2006.04.001.
[5]	J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states, Ⅱ, J. Differential Equations, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.
[6]	D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.
[7]	E. N. Dancer, F. Gladiali and M. Grossi, On the Hardy-Sobolev equation, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 299-336. doi: 10.1017/S0308210516000135.
[8]	E. N. Dancer and J. C. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscripta Math., 123 (2007), 493-511. doi: 10.1007/s00229-007-0110-6.
[9]	F. Gladiali, M. Grossi and S. L. 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.
[10]	F. Gladiali, M. Grossi, F. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differential Equations, 40 (2011), 295-317. doi: 10.1007/s00526-010-0341-3.
[11]	M. Hénon, Numerical Experiments on The Stability of Spherical Stellar Systems, Astronomy and Astrophysics, 1973.
[12]	H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Mathematical Sciences, Springer-Verlag, New York, 2004. doi: 10.1007/b97365.
[13]	H. Kielhöfer, A bifurcation theorem for potential operators, J. Funct. Anal., 77 (1988), 1-8. doi: 10.1016/0022-1236(88)90073-0.
[14]	Z. Lou, T. Weth and Z. Zhang, Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type, Z. Angew. Math. Phys., 70 (2019), Art. 35, 19 pp, arXiv: 1803.02712. doi: 10.1007/s00033-019-1080-8.
[15]	E. Moreira dos Santos and F. Pacella, Morse index of radial nodal solutions of Hénon type equations in dimension two, Commun. Contemp. Math., 19 (2017), 1650042, 16 pp. doi: 10.1142/S0219199716500425.
[16]	K. Nagasaki, Radial solutions for $\Delta u + \|x\|^l \|u\|^{p-1}u = 0$ on the unit ball in $\mathbb{R}^n$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 211-232.
[17]	Y. Naito, Bounded solutions with prescribed numbers of zeros for the Emden-Fowler differential equation, Hiroshima Math. J., 24 (1994), 177-220. doi: 10.32917/hmj/1206128140.
[18]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.
[19]	A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97. doi: 10.1007/s00209-006-0060-9.
[20]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326. doi: 10.1007/s00526-004-0302-9.
[21]	D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.
[22]	D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[23]	E. Yanagida, Structure of radial solutions to $\Delta u + K(\|x\|)\|u\|^{p-1}u=0$ in $\mathbb{R}^N$, SIAM J. Math. Anal., 27 (1996), 997-1014. doi: 10.1137/0527053.

[1]	Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[2]	Anna Lisa Amadori. Global bifurcation for the Hénon problem. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4797-4816. doi: 10.3934/cpaa.2020212
[3]	Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081
[4]	Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062
[5]	Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[6]	Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1559-1600. doi: 10.3934/cpaa.2021033
[7]	Craig Cowan, Abdolrahman Razani. Singular solutions of a Hénon equation involving a nonlinear gradient term. Communications on Pure and Applied Analysis, 2022, 21 (1) : 141-158. doi: 10.3934/cpaa.2021172
[8]	Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309
[9]	Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107
[10]	Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823
[11]	Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031
[12]	Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393
[13]	Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975
[14]	Josef DiblÍk, Rigoberto Medina. Exact asymptotics of positive solutions to Dickman equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 101-121. doi: 10.3934/dcdsb.2018007
[15]	Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[16]	Gustavo S. Costa, Giovany M. Figueiredo. Existence and concentration of nodal solutions for a subcritical p&q equation. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5077-5095. doi: 10.3934/cpaa.2020227
[17]	Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326
[18]	Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809
[19]	Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems and Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795
[20]	R. Estrada. Boundary layers and spectral content asymptotics. Conference Publications, 1998, 1998 (Special) : 242-252. doi: 10.3934/proc.1998.1998.242

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Joel Kübler Tobias Weth
on Google Scholar
Joel Kübler Tobias Weth

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Joel Kübler Tobias Weth
on Google Scholar
Joel Kübler Tobias Weth

[Back to Top]