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Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation
Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, D-60629 Frankfurt a.M., Germany |
$ \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*} $ |
$ {\bf B} \subset \mathbb{R}^N,N\geq 3 $ |
$ p>2 $ |
$ \alpha \to +\infty $ |
$ K $ |
$ C^1 $ |
$ u_\alpha $ |
$ (H) $ |
$ K $ |
$ u_\alpha(0)>0 $ |
$ \alpha \mapsto (\alpha,u_\alpha) $ |
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 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. |
[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. 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. |
[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 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. |
[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. 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. |
[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. |
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