×

The Hénon problem with large exponent in the disc. (English) Zbl 1437.35378

Summary: In this paper we consider the Hénon problem in the unit disc with Dirichlet boundary conditions. We study the asymptotic profile of least energy and nodal least energy radial solutions and then deduce the exact computation of their Morse index for large values of the exponent \(p\). As a consequence of this computation a multiplicity result for positive and nodal solutions is obtained.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adimurthi, M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Am. Math. Soc., 132, 1013-1019 (2004) · Zbl 1083.35035
[2] Aftalion, A.; Pacella, F., Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C. R. Acad. Sci. Paris, Ser. I, 339, 339-344 (2004) · Zbl 1113.35063
[3] Amadori, A. L., On the asymptotically linear Hénon problem (2019), Preprint
[4] Amadori, A. L., Global bifurcation for the Hénon problem (2019), Preprint
[5] Amadori, A. L.; Gladiali, F., Bifurcation and symmetry breaking for the Hénon equation, Adv. Differ. Equ., 19, 7-8, 755-782 (2014) · Zbl 1320.35056
[6] Amadori, A. L.; Gladiali, F., On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s (2018)
[7] Amadori, A. L.; Gladiali, F., On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s, part II (2018)
[8] Amadori, A. L.; Gladiali, F., Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem, Calc. Var., 58 (2019) · Zbl 1431.35044
[9] Castro, A.; Cossio, J.; Neuberger, J. M., A sign-changing solution for a superlinear Dirichlet problem, Rocky Mt. J. Math., 27, 1041-1053 (1997) · Zbl 0907.35050
[10] Bartsch, T.; Weth, T., A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22, 1-14 (2003) · Zbl 1094.35041
[11] Bartsch, T.; Weth, T.; Willem, M., Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96, 1-18 (2005) · Zbl 1206.35086
[12] Cowan, C.; Ghoussoub, N., Estimates on pull-in distances in microelectromechanical systems models and other nonlinear eigenvalue problems, SIAM J. Math. Anal., 42, 1949-1966 (2010) · Zbl 1219.35164
[13] De Marchis, F.; Ianni, I.; Pacella, F., Exact Morse index computation for nodal radial solutions of Lane-Emden problems, Math. Ann., 367, 1-2, 185-227 (2017) · Zbl 1379.35132
[14] De Marchis, F.; Ianni, I.; Pacella, F., A Morse index formula for radial solutions of Lane-Emden problems, Adv. Math., 322, 682-737 (2017) · Zbl 1393.35058
[15] Esposito, P.; Musso, M.; Pistoia, A., On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. Math. Soc., 94, 2, 497-519 (2007) · Zbl 1387.35219
[16] Esposito, P.; Pistoia, A.; Wei, J., Concentrating solutions for the Hénon equation in \(\mathbb{R}^2\), J. Anal. Math., 100, 1, 249-280 (2006) · Zbl 1173.35504
[17] Gladiali, F., A monotonicity result under symmetry and Morse index constraints in the plane (2019), Preprint
[18] Gladiali, F.; Grossi, M.; Neves, S. L.N., Nonradial solutions for the Hénon equation in \(\mathbb{R}^N\), Adv. Math., 249, 1-36 (2013) · Zbl 1335.35077
[19] Gladiali, F.; Grossi, M.; Neves, S. L.N., Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane, Commun. Contemp. Math., 18 (2016) · Zbl 1355.35080
[20] Gladiali, F.; Ianni, I., Quasiradial nodal solutions for the Lane-Emden problem in the ball (2017) · Zbl 1437.35336
[21] Grossi, M.; Grumiau, C.; Pacella, F., Lane-Emden problems: asymptotic behavior of low energy nodal solutions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 30, 121-140 (2013) · Zbl 1266.35106
[22] Grossi, M.; Grumiau, C.; Pacella, F., Lane Emden problems with large exponents and singular Liouville equations, J. Math. Pures Appl., 101, 6, 735-754 (2014) · Zbl 1292.35109
[23] Hénon, M., Numerical experiments on the stability oh spherical stellar systems, Astron. Astrophys., 24, 229-238 (1973)
[24] Kübler, J.; Weth, T., Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation, Discrete Contin. Dyn. Syst., Ser. A (2019)
[25] Lou, Z.; Weth, T.; Zhang, Z., Symmetry breaking via Morse index for equations and systems of Hénon-Schrodinger type, Z. Angew. Math. Phys., 70, 1, Article 35 pp. (2019) · Zbl 1412.35108
[26] Ni, W. M., A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31, 801-807 (1982) · Zbl 0515.35033
[27] Ni, W. M.; Nussbaum, R. D., Uniqueness and nonuniqueness for positive radial solutions of \(\operatorname{\Delta} u + f(u, r) = 0\), Commun. Pure Appl. Math., 38, 67-108 (1985) · Zbl 0581.35021
[28] Pacella, F.; Weth, T., Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Am. Math. Soc., 135, 1753-1762 (2007) · Zbl 1190.35096
[29] Prajapat, J.; Tarantello, G., On a class of elliptic problems in \(\mathbb{R}^2\): symmetry and uniqueness results, Proc. R. Soc. Edinb., 131A, 967-985 (2001) · Zbl 1009.35018
[30] Palais, R. S., The principle of symmetric criticality, Commun. Math. Phys., 69, 19-30 (1979) · Zbl 0417.58007
[31] Ren, X.; Wei, J., On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Am. Math. Soc., 343, 749-763 (1994) · Zbl 0804.35042
[32] Smets, D.; Willem, M.; Su, J., Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4, 467-480 (2002) · Zbl 1160.35415
[33] Zhang, Y.-B.; Yang, H.-T., Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity, Acta Math. Appl. Sin., 31, 1, 261-276 (2015) · Zbl 1316.35112
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.