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

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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
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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 and Continuous Dynamical Systems, 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, Springer-Verlag, Berlin, Heidelberg, New York, 1998.
[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), 591-631.
[3]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, J. Differential Equations, 163 (2000), 429-474.
[4]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, II, J. Differential Equations, 173 (2001), 321-355.
[5]	J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity, J. Differential Equations, 244 (2008), 2473-2497.
[6]	J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. and Partial Differential Equations, 24 (2005), 459-477.
[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), 803-828. doi: 10.1016/j.anihpc.2006.04.001.
[8]	J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states II, J. Differential Equations, 216 (2005), 78-108.
[9]	D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.
[10]	D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.
[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), 281-302. doi: 10.1016/j.anihpc.2006.09.003.
[12]	G. Chen, W. M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear ellptic equations, Inter. Jour. Bifur. Chaos, 10 (2000), 1565-1612. doi: 10.1142/S0218127400001006.
[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), 883-898. doi: 10.1512/iumj.1999.48.1596.
[14]	G. F. D. Duff, Partial differential equations, in "Mathematical Expositions," University of Toronto Press, Toronto, 1956.
[15]	P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.
[16]	B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.
[17]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2^nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983.
[18]	M. Gruter and K. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342. doi: 10.1007/BF01166225.
[19]	N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations, 247 (2009), 1311-1333.
[20]	M. Hénon, Numerical experiments on the spherical stellar systems, Astronomy and Astrophysics, 24 (1973), 229-238.
[21]	B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Mathematics, 1150, Springer-Verlag, Berlin, Heidelberg, New York and Tokyo, 1985.
[22]	Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490. doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z.
[23]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31(6) (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.
[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), 819-851. doi: 10.1002/cpa.3160440705.
[25]	W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[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), 1-20. doi: 10.1215/S0012-7094-92-06701-9.
[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), 731-768. doi: 10.1002/cpa.3160480704.
[28]	A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97. doi: 10.1007/s00209-006-0060-9.
[29]	S. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0.$, Soviet Math. Dokl., 6 (1965), 1408-1411.
[30]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1984.
[31]	O. Rey, Boundary effect for an elliptic Neumann problem with critical nonlinearity, Comm. in Partial Differential Equations, 22 (1997), 1055-1139.
[32]	S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disk, Commun. Contemp. Math., 8 (2006), 823-839. doi: 10.1142/S0219199706002295.
[33]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326.
[34]	M. Struwe, "Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, 1990.
[35]	D. Smets, J. Su and M. Willem, Nonradial ground states for the Hénon equation, Communications in Contemporary Mathematics, 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[36]	N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, conference on harmonic analysis in honor of Antoni Zygmund, Wadsworth Math. Ser., Belmont, CA, (1983), 821-837.
[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), 350-378. doi: 10.1016/j.matpur.2007.07.001.

show all references

References:

[1]	T. Aubin, "Some Nonlinear Problems In Riemannian Geometry," Springer Monographs in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1998.
[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), 591-631.
[3]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, J. Differential Equations, 163 (2000), 429-474.
[4]	J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, II, J. Differential Equations, 173 (2001), 321-355.
[5]	J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity, J. Differential Equations, 244 (2008), 2473-2497.
[6]	J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. and Partial Differential Equations, 24 (2005), 459-477.
[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), 803-828. doi: 10.1016/j.anihpc.2006.04.001.
[8]	J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states II, J. Differential Equations, 216 (2005), 78-108.
[9]	D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.
[10]	D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.
[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), 281-302. doi: 10.1016/j.anihpc.2006.09.003.
[12]	G. Chen, W. M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear ellptic equations, Inter. Jour. Bifur. Chaos, 10 (2000), 1565-1612. doi: 10.1142/S0218127400001006.
[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), 883-898. doi: 10.1512/iumj.1999.48.1596.
[14]	G. F. D. Duff, Partial differential equations, in "Mathematical Expositions," University of Toronto Press, Toronto, 1956.
[15]	P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.
[16]	B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.
[17]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2^nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983.
[18]	M. Gruter and K. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342. doi: 10.1007/BF01166225.
[19]	N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations, 247 (2009), 1311-1333.
[20]	M. Hénon, Numerical experiments on the spherical stellar systems, Astronomy and Astrophysics, 24 (1973), 229-238.
[21]	B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Mathematics, 1150, Springer-Verlag, Berlin, Heidelberg, New York and Tokyo, 1985.
[22]	Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490. doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z.
[23]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31(6) (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.
[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), 819-851. doi: 10.1002/cpa.3160440705.
[25]	W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[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), 1-20. doi: 10.1215/S0012-7094-92-06701-9.
[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), 731-768. doi: 10.1002/cpa.3160480704.
[28]	A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97. doi: 10.1007/s00209-006-0060-9.
[29]	S. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0.$, Soviet Math. Dokl., 6 (1965), 1408-1411.
[30]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1984.
[31]	O. Rey, Boundary effect for an elliptic Neumann problem with critical nonlinearity, Comm. in Partial Differential Equations, 22 (1997), 1055-1139.
[32]	S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disk, Commun. Contemp. Math., 8 (2006), 823-839. doi: 10.1142/S0219199706002295.
[33]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326.
[34]	M. Struwe, "Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, 1990.
[35]	D. Smets, J. Su and M. Willem, Nonradial ground states for the Hénon equation, Communications in Contemporary Mathematics, 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[36]	N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, conference on harmonic analysis in honor of Antoni Zygmund, Wadsworth Math. Ser., Belmont, CA, (1983), 821-837.
[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), 350-378. doi: 10.1016/j.matpur.2007.07.001.

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