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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

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