American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter
  • CPAA Home
  • This Issue
  • Next Article
    Multiplicity results for a class of elliptic problems with nonlinear boundary condition
March  2013, 12(2): 803-813. doi: 10.3934/cpaa.2013.12.803

Some results on two-dimensional Hénon equation with large exponent in nonlinearity

Chunyi Zhao 1,

1. 

Department of Mathematics, East China Normal University, Shanghai 200241, China

Received  September 2011 Revised  December 2011 Published  September 2012

The Hénon equation on a bounded domain in $R^2$ with large exponent in the nonlinear term is studied in this paper. We investigate positive solution obtained by the variational method and give its asymptotic behavior as the nonlinear exponent gets large.
Keywords: least energy solution, Hénon equation, non-radial symmetry..
Mathematics Subject Classification: Primary: 35J25; Secondary: 35B4.
Citation: Chunyi Zhao. Some results on two-dimensional Hénon equation with large exponent in nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 803-813. doi: 10.3934/cpaa.2013.12.803
References:
[1]

Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2004), 1013-1019 (electronic). doi: 10.1090/S0002-9939-03-07301-5.  Google Scholar

[2]

V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728. doi: 10.1016/j.jmaa.2007.10.052.  Google Scholar

[3]

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

[4]

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

[5]

W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-623. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

G. Chen, W.-M. Ni and J.X. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1565-1612. doi: 10.1142/S0218127400001006.  Google Scholar

[7]

D. M. Cao and S. J. 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

[8]

D. M. Cao, S. J. Peng and S. S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.  Google Scholar

[9]

M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525. doi: 10.1016/j.jde.2008.06.018.  Google Scholar

[10]

P. Esposito, A. Pistoia and J. C. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.  Google Scholar

[11]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[12]

M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 281-302. doi: 10.1016/j.anihpc.2006.09.003.  Google Scholar

[13]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238. Google Scholar

[14]

S. J. Li and S. J. Peng, Asymptotic behavior on the Hénon equation with supercritical exponent, Sci. China Ser. A, 52 (2009), 2185-2194. doi: 10.1007/s11425-009-0094-7.  Google Scholar

[15]

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

[16]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. doi: 10.1002/cpa.3160440705.  Google Scholar

[17]

S. J. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137-162. doi: 10.1007/s10255-005-0293-0.  Google Scholar

[18]

J. Prajapat and G. Tarantello, On a class of elliptic problem in $\mathbb R^2$: symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967-985. doi: 10.1017/S0308210500001219.  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]

X. F. Ren and J. C. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763. doi: 10.1090/S0002-9947-1994-1232190-7.  Google Scholar

[21]

X. F. Ren and J. C. Wei, Single-point condensation and least-energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120. doi: 10.1090/S0002-9939-96-03156-5.  Google Scholar

[22]

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

[23]

D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.  Google Scholar

[24]

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

show all references

References:
[1]

Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2004), 1013-1019 (electronic). doi: 10.1090/S0002-9939-03-07301-5.  Google Scholar

[2]

V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728. doi: 10.1016/j.jmaa.2007.10.052.  Google Scholar

[3]

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

[4]

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

[5]

W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-623. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

G. Chen, W.-M. Ni and J.X. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1565-1612. doi: 10.1142/S0218127400001006.  Google Scholar

[7]

D. M. Cao and S. J. 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

[8]

D. M. Cao, S. J. Peng and S. S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.  Google Scholar

[9]

M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525. doi: 10.1016/j.jde.2008.06.018.  Google Scholar

[10]

P. Esposito, A. Pistoia and J. C. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.  Google Scholar

[11]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[12]

M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 281-302. doi: 10.1016/j.anihpc.2006.09.003.  Google Scholar

[13]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238. Google Scholar

[14]

S. J. Li and S. J. Peng, Asymptotic behavior on the Hénon equation with supercritical exponent, Sci. China Ser. A, 52 (2009), 2185-2194. doi: 10.1007/s11425-009-0094-7.  Google Scholar

[15]

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

[16]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. doi: 10.1002/cpa.3160440705.  Google Scholar

[17]

S. J. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137-162. doi: 10.1007/s10255-005-0293-0.  Google Scholar

[18]

J. Prajapat and G. Tarantello, On a class of elliptic problem in $\mathbb R^2$: symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967-985. doi: 10.1017/S0308210500001219.  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]

X. F. Ren and J. C. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763. doi: 10.1090/S0002-9947-1994-1232190-7.  Google Scholar

[21]

X. F. Ren and J. C. Wei, Single-point condensation and least-energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120. doi: 10.1090/S0002-9939-96-03156-5.  Google Scholar

[22]

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

[23]

D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.  Google Scholar

[24]

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

[1]

Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055
[2]

Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
[3]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[4]

Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006
[5]

Shun Kodama. A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential. Communications on Pure & Applied Analysis, 2017, 16 (2) : 671-698. doi: 10.3934/cpaa.2017033
[6]

Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099
[7]

Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
[8]

Eudes. M. Barboza, Olimpio H. Miyagaki, Fábio R. Pereira, Cláudia R. Santana. Radial solutions for a class of Hénon type systems with partial interference with the spectrum. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3159-3187. doi: 10.3934/cpaa.2020137
[9]

Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083
[10]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190
[11]

Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915
[12]

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1559-1600. doi: 10.3934/cpaa.2021033
[13]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Hénon equation involving a nonlinear gradient term. Communications on Pure & Applied Analysis, 2022, 21 (1) : 141-158. doi: 10.3934/cpaa.2021172
[14]

Anna Lisa Amadori. Global bifurcation for the Hénon problem. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4797-4816. doi: 10.3934/cpaa.2020212
[15]

Adnan H. Sabuwala, Doreen De Leon. Particular solution to the Euler-Cauchy equation with polynomial non-homegeneities. Conference Publications, 2011, 2011 (Special) : 1271-1278. doi: 10.3934/proc.2011.2011.1271
[16]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[17]

Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control & Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1
[18]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[19]

Shoichi Hasegawa. A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1189-1198. doi: 10.3934/cpaa.2017058
[20]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy