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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

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 & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
References:
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A. Amadori and F. Gladiali, Bifurcation and symmetry breaking for the Hénon equation, Adv. Differential Equations, 19 (2014), 755-782.   Google Scholar

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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 Google Scholar

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A. Amadori and F. Gladiali, Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem, preprint, arXiv: 1810.11046 Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Hénon, Numerical Experiments on The Stability of Spherical Stellar Systems, Astronomy and Astrophysics, 1973. Google Scholar

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H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Mathematical Sciences, Springer-Verlag, New York, 2004. doi: 10.1007/b97365.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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 Google Scholar

[3]

A. Amadori and F. Gladiali, Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem, preprint, arXiv: 1810.11046 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Hénon, Numerical Experiments on The Stability of Spherical Stellar Systems, Astronomy and Astrophysics, 1973. Google Scholar

[12]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Mathematical Sciences, Springer-Verlag, New York, 2004. doi: 10.1007/b97365.  Google Scholar

[13]

H. Kielhöfer, A bifurcation theorem for potential operators, J. Funct. Anal., 77 (1988), 1-8.  doi: 10.1016/0022-1236(88)90073-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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