Article Contents
Article Contents

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

• 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 $p\to \frac{2N}{N-2}$.
Mathematics Subject Classification: Primary: 35J20, 35J61.

 Citation:

•  [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.