On the location of a peak point of a least energy solution for Hénon equation

Jaeyoung Byeon; Sungwon Cho; Junsang Park

doi:10.3934/dcds.2011.30.1055

Previous Article
Radial symmetry of solutions for some integral systems of Wolff type
DCDS Home
This Issue
Next Article
Asymptotics for a generalized Cahn-Hilliard equation with forcing terms

November 2011, 30(4): 1055-1081. doi: 10.3934/dcds.2011.30.1055

On the location of a peak point of a least energy solution for Hénon equation

Jaeyoung Byeon ^1, , Sungwon Cho ^2, and Junsang Park ^3,

1.	Department of Mathematics & PMI, POSTECH, Pohang, Kyungbuk 790-784
2.	Department of Mathematics Education, Gwangju National University of Education, 93 Pilmunlo Bugku, Gwangju 500-703
3.	Department of Mathematics, POSTECH, Pohang, Kyungbuk 790-784

Received May 2010 Revised February 2011 Published May 2011

Abstract
Related Papers

Let $\Omega $ be a smooth bounded domain. We are concerned about the following nonlinear elliptic problem:

$\Delta u + |x|^{\alpha}u^{p} = 0, \ u > 0 \quad$ in $\Omega$,
$\ u = 0 \quad$ on $\partial \Omega $,

where $\alpha > 0, p \in (1,\frac{n+2}{n-2}).$ In this paper, we show that for $n \ge 8$, a maximum point $x_{\alpha }$ of a least energy solution of above problem converges to a point $x_0 \in \partial^$*$ \Omega $ satisfying $H(x_0) = \min_$_{$\w \in \partial ^$*$ \Omega$}$H( w )$ as $\alpha \to \infty,$ where $H$ is the mean curvature on $\partial \Omega $ and $\partial ^$*$\Omega \equiv \{ x \in \partial \Omega : |x| \ge |y|$ for any $y \in \Omega \}.$

Keywords: asymptotic profile, mean curvature., Hénon equation, least energy solutions.

Mathematics Subject Classification: Primary: 35J60, 35J20; Secondary: 35J25, 47J3.

Citation: Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055

References:

[1]	T. Aubin, "Some Nonlinear Problems In Riemannian Geometry,", Springer Monographs in Mathematics, (1998). Google Scholar
[2]	Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent,, Comm. in Partial Differential Equations, 20 (1995), 591. Google Scholar
[3]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations,, J. Differential Equations, 163 (2000), 429. Google Scholar
[4]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, II,, J. Differential Equations, 173 (2001), 321. Google Scholar
[5]	J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473. Google Scholar
[6]	J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds,, Calc. Var. and Partial Differential Equations, 24 (2005), 459. Google Scholar
[7]	J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states,, Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006), 803. doi: 10.1016/j.anihpc.2006.04.001. Google Scholar
[8]	J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states II,, J. Differential Equations, 216 (2005), 78. Google Scholar
[9]	D. Cao and S. 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
[10]	D. Cao, S. Peng and 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
[11]	M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues,, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 281. doi: 10.1016/j.anihpc.2006.09.003. Google Scholar
[12]	G. Chen, W. M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear ellptic equations,, Inter. Jour. Bifur. Chaos, 10 (2000), 1565. doi: 10.1142/S0218127400001006. Google Scholar
[13]	M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: 10.1512/iumj.1999.48.1596. Google Scholar
[14]	G. F. D. Duff, Partial differential equations,, in, (1956). Google Scholar
[15]	P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$,, J. Anal. Math., 100 (2006), 249. doi: 10.1007/BF02916763. Google Scholar
[16]	B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. Google Scholar
[17]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2^nd edition, (1983). Google Scholar
[18]	M. Gruter and K. Widman, The Green function for uniformly elliptic equations,, Manuscripta Math., 37 (1982), 303. doi: 10.1007/BF01166225. Google Scholar
[19]	N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms,, J. Differential Equations, 247 (2009), 1311. Google Scholar
[20]	M. Hénon, Numerical experiments on the spherical stellar systems,, Astronomy and Astrophysics, 24 (1973), 229. Google Scholar
[21]	B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Mathematics, 1150 (1985). Google Scholar
[22]	Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445. doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. Google Scholar
[23]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,, Indiana Univ. Math. J., 31(6) (1982), 801. doi: 10.1512/iumj.1982.31.31056. Google Scholar
[24]	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
[25]	W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247. doi: 10.1215/S0012-7094-93-07004-4. Google Scholar
[26]	W. M. Ni, X. B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1. doi: 10.1215/S0012-7094-92-06701-9. Google Scholar
[27]	W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 48 (1995), 731. doi: 10.1002/cpa.3160480704. Google Scholar
[28]	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
[29]	S. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0.$,, Soviet Math. Dokl., 6 (1965), 1408. Google Scholar
[30]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984). Google Scholar
[31]	O. Rey, Boundary effect for an elliptic Neumann problem with critical nonlinearity,, Comm. in Partial Differential Equations, 22 (1997), 1055. Google Scholar
[32]	S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disk,, Commun. Contemp. Math., 8 (2006), 823. doi: 10.1142/S0219199706002295. Google Scholar
[33]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth,, Calc. Var. Partial Differential Equations, 23 (2005), 301. Google Scholar
[34]	M. Struwe, "Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer-Verlag, (1990). Google Scholar
[35]	D. Smets, J. Su and M. Willem, Nonradial ground states for the Hénon equation,, Communications in Contemporary Mathematics, 4 (2002), 467. doi: 10.1142/S0219199702000725. Google Scholar
[36]	N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, conference on harmonic analysis in honor of Antoni Zygmund,, Wadsworth Math. Ser., (1983), 821. Google Scholar
[37]	J. Wei and S. Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth,, J. Math. Pures Appl., 88 (2007), 350. doi: 10.1016/j.matpur.2007.07.001. Google Scholar

show all references

References:

[1]	T. Aubin, "Some Nonlinear Problems In Riemannian Geometry,", Springer Monographs in Mathematics, (1998). Google Scholar
[2]	Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent,, Comm. in Partial Differential Equations, 20 (1995), 591. Google Scholar
[3]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations,, J. Differential Equations, 163 (2000), 429. Google Scholar
[4]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, II,, J. Differential Equations, 173 (2001), 321. Google Scholar
[5]	J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473. Google Scholar
[6]	J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds,, Calc. Var. and Partial Differential Equations, 24 (2005), 459. Google Scholar
[7]	J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states,, Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006), 803. doi: 10.1016/j.anihpc.2006.04.001. Google Scholar
[8]	J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states II,, J. Differential Equations, 216 (2005), 78. Google Scholar
[9]	D. Cao and S. 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
[10]	D. Cao, S. Peng and 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
[11]	M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues,, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 281. doi: 10.1016/j.anihpc.2006.09.003. Google Scholar
[12]	G. Chen, W. M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear ellptic equations,, Inter. Jour. Bifur. Chaos, 10 (2000), 1565. doi: 10.1142/S0218127400001006. Google Scholar
[13]	M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: 10.1512/iumj.1999.48.1596. Google Scholar
[14]	G. F. D. Duff, Partial differential equations,, in, (1956). Google Scholar
[15]	P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$,, J. Anal. Math., 100 (2006), 249. doi: 10.1007/BF02916763. Google Scholar
[16]	B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. Google Scholar
[17]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2^nd edition, (1983). Google Scholar
[18]	M. Gruter and K. Widman, The Green function for uniformly elliptic equations,, Manuscripta Math., 37 (1982), 303. doi: 10.1007/BF01166225. Google Scholar
[19]	N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms,, J. Differential Equations, 247 (2009), 1311. Google Scholar
[20]	M. Hénon, Numerical experiments on the spherical stellar systems,, Astronomy and Astrophysics, 24 (1973), 229. Google Scholar
[21]	B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Mathematics, 1150 (1985). Google Scholar
[22]	Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445. doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. Google Scholar
[23]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,, Indiana Univ. Math. J., 31(6) (1982), 801. doi: 10.1512/iumj.1982.31.31056. Google Scholar
[24]	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
[25]	W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247. doi: 10.1215/S0012-7094-93-07004-4. Google Scholar
[26]	W. M. Ni, X. B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1. doi: 10.1215/S0012-7094-92-06701-9. Google Scholar
[27]	W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 48 (1995), 731. doi: 10.1002/cpa.3160480704. Google Scholar
[28]	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
[29]	S. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0.$,, Soviet Math. Dokl., 6 (1965), 1408. Google Scholar
[30]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984). Google Scholar
[31]	O. Rey, Boundary effect for an elliptic Neumann problem with critical nonlinearity,, Comm. in Partial Differential Equations, 22 (1997), 1055. Google Scholar
[32]	S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disk,, Commun. Contemp. Math., 8 (2006), 823. doi: 10.1142/S0219199706002295. Google Scholar
[33]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth,, Calc. Var. Partial Differential Equations, 23 (2005), 301. Google Scholar
[34]	M. Struwe, "Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer-Verlag, (1990). Google Scholar
[35]	D. Smets, J. Su and M. Willem, Nonradial ground states for the Hénon equation,, Communications in Contemporary Mathematics, 4 (2002), 467. doi: 10.1142/S0219199702000725. Google Scholar
[36]	N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, conference on harmonic analysis in honor of Antoni Zygmund,, Wadsworth Math. Ser., (1983), 821. Google Scholar
[37]	J. Wei and S. Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth,, J. Math. Pures Appl., 88 (2007), 350. doi: 10.1016/j.matpur.2007.07.001. Google Scholar

[1]	Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
[2]	Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083
[3]	Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[4]	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
[5]	Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463
[6]	Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297
[7]	Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[8]	Yu-Zhu Wang, Si Chen, Menglong Su. Asymptotic profile of solutions to the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$. Evolution Equations & Control Theory, 2017, 6 (4) : 629-645. doi: 10.3934/eect.2017032
[9]	Rafał Celiński, Andrzej Raczyński. Asymptotic profile of solutions to a certain chemotaxis system. Communications on Pure & Applied Analysis, 2020, 19 (2) : 911-922. doi: 10.3934/cpaa.2020041
[10]	Jun Wang, Wei Wei, Jinju Xu. Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3243-3265. doi: 10.3934/cpaa.2019146
[11]	Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441
[12]	Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1
[13]	Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256
[14]	Qiuping Lu, Zhi Ling. Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2411-2429. doi: 10.3934/cpaa.2015.14.2411
[15]	Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
[16]	Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024
[17]	Luigi Montoro. On the shape of the least-energy solutions to some singularly perturbed mixed problems. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731
[18]	Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126
[19]	Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[20]	Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences

On the location of a peak point of a least energy solution for Hénon equation

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

On the location of a peak point of a least energy solution for Hénon equation

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors