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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 & 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.  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.  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. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 803.  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.  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.  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.  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.  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.  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.  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.  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.  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. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 281.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 803.  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.  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.  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.  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.  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.  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.  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.  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.  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. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 281.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1007/s00526-002-0180-y.  Google Scholar

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