# 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 & 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.  Google Scholar [2] A. L. Amadori, On the asymptotically linear Hénon problem, to appear, Commun. Contemp. Math., 2019. doi: 10.1142/S021919972050042X.  Google Scholar [3] A. L. Amadori and F. Gladiali, Bifurcation and symmetry breaking for the Hénon equation, Adv. Differ. Equ., 19 (2014), 755-782.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, preprint, arXiv: 1908.10503. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [2] A. L. Amadori, On the asymptotically linear Hénon problem, to appear, Commun. Contemp. Math., 2019. doi: 10.1142/S021919972050042X.  Google Scholar [3] A. L. Amadori and F. Gladiali, Bifurcation and symmetry breaking for the Hénon equation, Adv. Differ. Equ., 19 (2014), 755-782.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, preprint, arXiv: 1908.10503. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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