# American Institute of Mathematical Sciences

October  2020, 19(10): 4797-4816. doi: 10.3934/cpaa.2020212

## Global bifurcation for the Hénon problem

 Dipartimento di Scienze Applicate, Università di Napoli "Parthenope", Centro Direzionale di Napoli, Isola C4, 80143 Napoli, Italy

Received  September 2019 Revised  May 2020 Published  July 2020

Fund Project: The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

We prove the existence of nonradial solutions for the Hénon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $\alpha$. For sign-changing solutions, the case $\alpha = 0$ -Lane-Emden equation- is included. The obtained solutions form global continua which branch off from the curve of radial solutions $p\mapsto u_p$, and the number of branching points increases with both the number of nodal zones and the exponent $\alpha$. The proof technique relies on the index of fixed points in cones and provides information on the symmetry properties of the bifurcating solutions and the possible intersection and/or overlapping between different branches, thus allowing to separate them in some cases.

Citation: 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
##### References:
 [1] A. L. Amadori and F. Gladiali, The Hénon problem with large exponent in the disc, J. Differ. Equ., 268 (2020), 5892-5944.  doi: 10.1016/j.jde.2019.11.017. [2] A. L. Amadori, On the asymptotically linear Hénon problem, to appear, Commun. Contemp. Math., 2019. doi: 10.1142/S021919972050042X. [3] A. L. Amadori and F. Gladiali, Bifurcation and symmetry breaking for the Hénon equation, Adv. Differ. Equ., 19 (2014), 755-782. [4] A. L. Amadori and F. Gladiali, Nonradial sign changing solutions to Lane-Emden problem in an annulus, Nonlinear Anal., 155 (2017), 294-305.  doi: 10.1016/j.na.2017.02.027. [5] A. L. Amadori and F. Gladiali, Asymptotic profile and morse index of nodal radial solutions to the Hénon problem, Calc. Var. Partial Differ. Equ., 58 (2019), 1-47.  doi: 10.1007/s00526-019-1606-0. [6] A. L. Amadori and F. Gladiali, On a singular eigenvalue problem and its applications in computing the morse index of solutions to semilinear PDE's: II, Nonlinearity, 33 (2020), 2541-2561.  doi: 10.1088/1361-6544/ab7639. [7] A. L. Amadori and F. Gladiali, On a singular eigenvalue problem and its applications in computing the morse index of solutions to semilinear PDE's, Nonlinear Anal. Real World Appl., 55 (2020), Art. 103133. doi: 10.1016/j.nonrwa.2020.103133. [8] M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.  doi: 10.1515/ans-2004-0406. [9] T. Bartsch, T. D'Aprile and A. Pistoia, On the profile of sign-changing solutions of an almost critical problem in the ball, Bull. Lond. Math. Soc., 45 (2013), 1246-1258.  doi: 10.1112/blms/bdt061. [10] T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on ${\mathbb R}^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.  doi: 10.1007/BF00953069. [11] E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7. [12] E. N. Dancer and P. Hess, Global breaking of symmetry of positive solutions on two-dimensional annuli, Differ. Integral Equ., 5 (1992), 903-913. [13] E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscr. Math., 123 (2007), 493-511.  doi: 10.1007/s00229-007-0110-6. [14] F. De Marchis, I. Ianni and F. Pacella, A Morse index formula for radial solutions of LaneEmden problems, Adv. Math., 322 (2017), 682-737.  doi: 10.1016/j.aim.2017.10.026. [15] F. De Marchis, I. Ianni and F. Pacella, Exact Morse index computation for nodal radial solutions of Lane-Emden problems, Math. Ann., 367 (2017), 185-227.  doi: 10.1007/s00208-016-1381-6. [16] P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in ${\mathbb R}^2$, J. Anal. Math., 100 (2006), 249-280.  doi: 10.1007/BF02916763. [17] P. Figueroa and S. L. N. Neves, Nonradial solutions for the Hénon equation close to the threshold, Adv. Nonlinear Stud., 19 (2019), 757-770.  doi: 10.1515/ans-2019-2052. [18] F. Gladiali, A global bifurcation result for a semilinear elliptic equation, J. Math. Anal. Appl., 369 (2010), 306-311.  doi: 10.1016/j.jmaa.2010.03.018. [19] F. Gladiali and I. Ianni, Quasi-radial solutions for the Lane-Emden problem in the ball, NoDea Nonlinear Differ. Equ. Appl., 27 (2020), Art. 13. doi: 10.1007/s00030-020-0616-0. [20] I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, preprint, arXiv: 1908.10503. [21] J. Kubler and T. Weth, Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation, Discrete Contin. Dyn. Syst., 40 (2019), Art. 3629. doi: 10.3934/dcds.2020032. [22] W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u + f(u, r) = 0$, Commun. Pure Appl. Math., 38 (1985), Art. 67. doi: 10.1002/cpa.3160380105. [23] A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), Art. 75. doi: 10.1007/s00209-006-0060-9. [24] M. Willem, D. Smets and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), Art. 467. doi: 10.1142/S0219199702000725. [25] Y. B. Zhang and H. T. Yang, Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity, Acta Math. Appl. Sin., 31 (2015), 261-276.  doi: 10.1007/s10255-015-0465-5.

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##### References:
 [1] A. L. Amadori and F. Gladiali, The Hénon problem with large exponent in the disc, J. Differ. Equ., 268 (2020), 5892-5944.  doi: 10.1016/j.jde.2019.11.017. [2] A. L. Amadori, On the asymptotically linear Hénon problem, to appear, Commun. Contemp. Math., 2019. doi: 10.1142/S021919972050042X. [3] A. L. Amadori and F. Gladiali, Bifurcation and symmetry breaking for the Hénon equation, Adv. Differ. Equ., 19 (2014), 755-782. [4] A. L. Amadori and F. Gladiali, Nonradial sign changing solutions to Lane-Emden problem in an annulus, Nonlinear Anal., 155 (2017), 294-305.  doi: 10.1016/j.na.2017.02.027. [5] A. L. Amadori and F. Gladiali, Asymptotic profile and morse index of nodal radial solutions to the Hénon problem, Calc. Var. Partial Differ. Equ., 58 (2019), 1-47.  doi: 10.1007/s00526-019-1606-0. [6] A. L. Amadori and F. Gladiali, On a singular eigenvalue problem and its applications in computing the morse index of solutions to semilinear PDE's: II, Nonlinearity, 33 (2020), 2541-2561.  doi: 10.1088/1361-6544/ab7639. [7] A. L. Amadori and F. Gladiali, On a singular eigenvalue problem and its applications in computing the morse index of solutions to semilinear PDE's, Nonlinear Anal. Real World Appl., 55 (2020), Art. 103133. doi: 10.1016/j.nonrwa.2020.103133. [8] M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.  doi: 10.1515/ans-2004-0406. [9] T. Bartsch, T. D'Aprile and A. Pistoia, On the profile of sign-changing solutions of an almost critical problem in the ball, Bull. Lond. Math. Soc., 45 (2013), 1246-1258.  doi: 10.1112/blms/bdt061. [10] T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on ${\mathbb R}^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.  doi: 10.1007/BF00953069. [11] E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7. [12] E. N. Dancer and P. Hess, Global breaking of symmetry of positive solutions on two-dimensional annuli, Differ. Integral Equ., 5 (1992), 903-913. [13] E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscr. Math., 123 (2007), 493-511.  doi: 10.1007/s00229-007-0110-6. [14] F. De Marchis, I. Ianni and F. Pacella, A Morse index formula for radial solutions of LaneEmden problems, Adv. Math., 322 (2017), 682-737.  doi: 10.1016/j.aim.2017.10.026. [15] F. De Marchis, I. Ianni and F. Pacella, Exact Morse index computation for nodal radial solutions of Lane-Emden problems, Math. Ann., 367 (2017), 185-227.  doi: 10.1007/s00208-016-1381-6. [16] P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in ${\mathbb R}^2$, J. Anal. Math., 100 (2006), 249-280.  doi: 10.1007/BF02916763. [17] P. Figueroa and S. L. N. Neves, Nonradial solutions for the Hénon equation close to the threshold, Adv. Nonlinear Stud., 19 (2019), 757-770.  doi: 10.1515/ans-2019-2052. [18] F. Gladiali, A global bifurcation result for a semilinear elliptic equation, J. Math. Anal. Appl., 369 (2010), 306-311.  doi: 10.1016/j.jmaa.2010.03.018. [19] F. Gladiali and I. Ianni, Quasi-radial solutions for the Lane-Emden problem in the ball, NoDea Nonlinear Differ. Equ. Appl., 27 (2020), Art. 13. doi: 10.1007/s00030-020-0616-0. [20] I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, preprint, arXiv: 1908.10503. [21] J. Kubler and T. Weth, Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation, Discrete Contin. Dyn. Syst., 40 (2019), Art. 3629. doi: 10.3934/dcds.2020032. [22] W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u + f(u, r) = 0$, Commun. Pure Appl. Math., 38 (1985), Art. 67. doi: 10.1002/cpa.3160380105. [23] A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), Art. 75. doi: 10.1007/s00209-006-0060-9. [24] M. Willem, D. Smets and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), Art. 467. doi: 10.1142/S0219199702000725. [25] Y. B. Zhang and H. T. Yang, Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity, Acta Math. Appl. Sin., 31 (2015), 261-276.  doi: 10.1007/s10255-015-0465-5.
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