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

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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

Abstract
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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 $p\to \frac{2N}{N-2}$.

Keywords: blow-up, Robin boundary condition, asymptotic behavior., Hénon equation.

Mathematics Subject Classification: Primary: 35J20, 35J6.

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

References:

[1]	Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi, Nonlinear Anal., Scuola Norm. Sup. Pisa, (1991), 9-25.
[2]	Adimurthi and G. Mancini, Geometry and topology of the boundary in the critical Neumann problem, J. Reine Angew. Math., 456 (1994), 1-18. doi: 10.1515/crll.1994.456.1.
[3]	Adimurthi and S. L. Yadava, Positive solution for Neumann problem with critical nonlinearity on boundary, Comm. Partial Differential Equations, 16 (1991), 1733-1760. doi: 10.1080/03605309108820821.
[4]	H. Brezis and E. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal., 62 (1985), 73-86. doi: 10.1016/0022-1236(85)90020-5.
[5]	J. Byeon and Z-Q. Wang, On the Hénon equation: Asymptotic profile of ground state I, Ann. I. H. Poincare., 23 (2006), 803-828. doi: 10.1016/j.anihpc.2006.04.001.
[6]	J. Byeon and Z-Q. Wang, On the Hénon equation: Asymptotic profile of ground state II, J. Differential Equation, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.
[7]	D. Cao and S. Peng, The asymptotic behavior 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]	Daomin Cao, E. S. Noussair and Shusen Yan, On a semilinear Robin prolem involving critical Sobolev exponent, Advanced Nonlinear Studies, 1 (2001), 43-77.
[9]	Yuxia Fu and Qiuyi Dai, Positive solutions of the Robin problem for semilinear elliptic equations on annuli, Rend. Lincei Mat. Appl., 19 (2008), 175-188. doi: 10.4171/RLM/516.
[10]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 8 (1981), 883-901. doi: 10.1080/03605308108820196.
[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), 389-318.
[12]	M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238. doi: 10.1007/978-94-010-9877-9_37.
[13]	Haiyang He, The Robin problem for the Hénon equation,, Accepted by Bulletin of the Australian Mathematic Society., ().
[14]	P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case, Rev. Mat. Iberoamericana., (1985), 145-201. doi: 10.4171/RMI/6.
[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), 819-851. doi: 10.1002/cpa.3160440705.
[16]	D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Henon equation, Comm. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[17]	D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problem, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.
[18]	X. J. Wang, Neumann problem for semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equation, 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.

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., Scuola Norm. Sup. Pisa, (1991), 9-25.
[2]	Adimurthi and G. Mancini, Geometry and topology of the boundary in the critical Neumann problem, J. Reine Angew. Math., 456 (1994), 1-18. doi: 10.1515/crll.1994.456.1.
[3]	Adimurthi and S. L. Yadava, Positive solution for Neumann problem with critical nonlinearity on boundary, Comm. Partial Differential Equations, 16 (1991), 1733-1760. doi: 10.1080/03605309108820821.
[4]	H. Brezis and E. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal., 62 (1985), 73-86. doi: 10.1016/0022-1236(85)90020-5.
[5]	J. Byeon and Z-Q. Wang, On the Hénon equation: Asymptotic profile of ground state I, Ann. I. H. Poincare., 23 (2006), 803-828. doi: 10.1016/j.anihpc.2006.04.001.
[6]	J. Byeon and Z-Q. Wang, On the Hénon equation: Asymptotic profile of ground state II, J. Differential Equation, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.
[7]	D. Cao and S. Peng, The asymptotic behavior 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]	Daomin Cao, E. S. Noussair and Shusen Yan, On a semilinear Robin prolem involving critical Sobolev exponent, Advanced Nonlinear Studies, 1 (2001), 43-77.
[9]	Yuxia Fu and Qiuyi Dai, Positive solutions of the Robin problem for semilinear elliptic equations on annuli, Rend. Lincei Mat. Appl., 19 (2008), 175-188. doi: 10.4171/RLM/516.
[10]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 8 (1981), 883-901. doi: 10.1080/03605308108820196.
[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), 389-318.
[12]	M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238. doi: 10.1007/978-94-010-9877-9_37.
[13]	Haiyang He, The Robin problem for the Hénon equation,, Accepted by Bulletin of the Australian Mathematic Society., ().
[14]	P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case, Rev. Mat. Iberoamericana., (1985), 145-201. doi: 10.4171/RMI/6.
[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), 819-851. doi: 10.1002/cpa.3160440705.
[16]	D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Henon equation, Comm. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[17]	D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problem, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.
[18]	X. J. Wang, Neumann problem for semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equation, 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.

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