American Institute of Mathematical Sciences

Advanced
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

Haiyang He 1,

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 $p\to \frac{2N}{N-2}$.
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 & 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.  Google Scholar

[2]

Adimurthi and G. Mancini, Geometry and topology of the boundary in the critical Neumann problem, J. Reine Angew. Math., 456 (1994), 1-18. 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), 1733-1760. doi: 10.1080/03605309108820821.  Google Scholar

[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.  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), 803-828. 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), 78-108. 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), 1-17. 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), 43-77.  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), 175-188. 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), 883-901. 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), 389-318. Google Scholar

[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.  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), 145-201. 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), 819-851. 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), 467-480. 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), 57-75. 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), 283-310. 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., Scuola Norm. Sup. Pisa, (1991), 9-25.  Google Scholar

[2]

Adimurthi and G. Mancini, Geometry and topology of the boundary in the critical Neumann problem, J. Reine Angew. Math., 456 (1994), 1-18. 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), 1733-1760. doi: 10.1080/03605309108820821.  Google Scholar

[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.  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), 803-828. 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), 78-108. 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), 1-17. 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), 43-77.  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), 175-188. 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), 883-901. 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), 389-318. Google Scholar

[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.  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), 145-201. 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), 819-851. 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), 467-480. 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), 57-75. 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), 283-310. doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar

[1]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[2]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[3]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[4]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[5]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006
[6]

Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure & Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435
[7]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[8]

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
[9]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861
[10]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[11]

C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523
[12]

Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113
[13]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[14]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108
[15]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032
[16]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[17]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697
[18]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[19]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271
[20]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy