November  2013, 12(6): 2393-2408. doi: 10.3934/cpaa.2013.12.2393

Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition

1. 

College of Mathematics and Computer Science, Key Laboratory of High Performance Computing, and Stochastic Information Processing(Ministry of Education of China), Hunan Normal University, Changsha, Hunan 410081, China

Received  April 2012 Revised  December 2012 Published  May 2012

In this paper, we consider the problem \begin{eqnarray} -\Delta u=|x|^\alpha u^{p-1}, x \in \Omega,\\ u>0, x \in \Omega,\\ \frac{\partial u}{\partial \nu }+\beta u=0, x\in \partial \Omega, \end{eqnarray} where $\Omega$ is the unit ball in $R^N$ centered at the origin with $N\geq 3$, $\alpha>0, \beta>\frac{N-2}{2}, p\geq 2$ and $\nu $ is the unit outward vector normal to $\partial \Omega$. We investigate the asymptotic behavior of the ground state solutions $u_p$ of (1) as $p\to \frac{2N}{N-2}$. We show that the ground state solutions $u_p$ has a unique maximum point $x_p\in \bar\Omega$. In addition, the ground state solutions is non-radial provided that $p\to \frac{2N}{N-2}$.
Citation: 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
References:
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Adimurthi and G. Mancini, Geometry and topology of the boundary in the critical Neumann problem,, J. Reine Angew. Math., 456 (1994), 1.  doi: 10.1515/crll.1994.456.1.  Google Scholar

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Adimurthi and S. L. Yadava, Positive solution for Neumann problem with critical nonlinearity on boundary,, Comm. Partial Differential Equations, 16 (1991), 1733.  doi: 10.1080/03605309108820821.  Google Scholar

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J. Byeon and Z-Q. Wang, On the Hénon equation: Asymptotic profile of ground state I,, Ann. I. H. Poincare., 23 (2006), 803.  doi: 10.1016/j.anihpc.2006.04.001.  Google Scholar

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D. Cao and S. Peng, The asymptotic behavior 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

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Daomin Cao, E. S. Noussair and Shusen Yan, On a semilinear Robin prolem involving critical Sobolev exponent,, Advanced Nonlinear Studies, 1 (2001), 43.   Google Scholar

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Yuxia Fu and Qiuyi Dai, Positive solutions of the Robin problem for semilinear elliptic equations on annuli,, Rend. Lincei Mat. Appl., 19 (2008), 175.  doi: 10.4171/RLM/516.  Google Scholar

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B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 8 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar

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Yonggen Gu and T. Liu, A priori estimate and existence of positive solutions of semilinear elliptic equations with the third boundary value problem,, J. Systems Sci. Complexity, 14 (2001), 389.   Google Scholar

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M. Hénon, Numerical experiments on the stability of spherical stellar systems,, Astronom. Astrophys., 24 (1973), 229.  doi: 10.1007/978-94-010-9877-9_37.  Google Scholar

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Haiyang He, The Robin problem for the Hénon equation,, Accepted by Bulletin of the Australian Mathematic Society., ().   Google Scholar

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P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case,, Rev. Mat. Iberoamericana., (1985), 145.  doi: 10.4171/RMI/6.  Google Scholar

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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

[16]

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

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D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problem,, Calc. Var. Partial Differential Equations, 18 (2003), 57.  doi: 10.1007/s00526-002-0180-y.  Google Scholar

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X. J. Wang, Neumann problem for semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equation, 93 (1991), 283.  doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar

show all references

References:
[1]

Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi,, Nonlinear Anal., (1991), 9.   Google Scholar

[2]

Adimurthi and G. Mancini, Geometry and topology of the boundary in the critical Neumann problem,, J. Reine Angew. Math., 456 (1994), 1.  doi: 10.1515/crll.1994.456.1.  Google Scholar

[3]

Adimurthi and S. L. Yadava, Positive solution for Neumann problem with critical nonlinearity on boundary,, Comm. Partial Differential Equations, 16 (1991), 1733.  doi: 10.1080/03605309108820821.  Google Scholar

[4]

H. Brezis and E. Lieb, Sobolev inequalities with remainder terms,, J. Funct. Anal., 62 (1985), 73.  doi: 10.1016/0022-1236(85)90020-5.  Google Scholar

[5]

J. Byeon and Z-Q. Wang, On the Hénon equation: Asymptotic profile of ground state I,, Ann. I. H. Poincare., 23 (2006), 803.  doi: 10.1016/j.anihpc.2006.04.001.  Google Scholar

[6]

J. Byeon and Z-Q. Wang, On the Hénon equation: Asymptotic profile of ground state II,, J. Differential Equation, 216 (2005), 78.  doi: 10.1016/j.jde.2005.02.018.  Google Scholar

[7]

D. Cao and S. Peng, The asymptotic behavior 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]

Daomin Cao, E. S. Noussair and Shusen Yan, On a semilinear Robin prolem involving critical Sobolev exponent,, Advanced Nonlinear Studies, 1 (2001), 43.   Google Scholar

[9]

Yuxia Fu and Qiuyi Dai, Positive solutions of the Robin problem for semilinear elliptic equations on annuli,, Rend. Lincei Mat. Appl., 19 (2008), 175.  doi: 10.4171/RLM/516.  Google Scholar

[10]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 8 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar

[11]

Yonggen Gu and T. Liu, A priori estimate and existence of positive solutions of semilinear elliptic equations with the third boundary value problem,, J. Systems Sci. Complexity, 14 (2001), 389.   Google Scholar

[12]

M. Hénon, Numerical experiments on the stability of spherical stellar systems,, Astronom. Astrophys., 24 (1973), 229.  doi: 10.1007/978-94-010-9877-9_37.  Google Scholar

[13]

Haiyang He, The Robin problem for the Hénon equation,, Accepted by Bulletin of the Australian Mathematic Society., ().   Google Scholar

[14]

P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case,, Rev. Mat. Iberoamericana., (1985), 145.  doi: 10.4171/RMI/6.  Google Scholar

[15]

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

[16]

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

[17]

D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problem,, Calc. Var. Partial Differential Equations, 18 (2003), 57.  doi: 10.1007/s00526-002-0180-y.  Google Scholar

[18]

X. J. Wang, Neumann problem for semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equation, 93 (1991), 283.  doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar

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