The Hénon equation with a critical exponent under the Neumann boundary condition

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September 2018, 38(9): 4353-4390. doi: 10.3934/dcds.2018190

The Hénon equation with a critical exponent under the Neumann boundary condition

Jaeyoung Byeon and Sangdon Jin

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Received August 2017 Revised April 2018 Published June 2018

For $n≥ 3$ and $p = (n+2)/(n-2), $ we consider the Hénon equation with the homogeneous Neumann boundary condition

$ -Δ u + u = |x|^{α}u^{p}, \; u > 0 \;\text{in} \; Ω,\ \ \frac{\partial u}{\partial ν} = 0 \; \text{ on }\;\partial Ω,$

where

$Ω \subset B(0,1) \subset \mathbb{R}^n, n ≥ 3$, $α≥ 0$ and $\partial^*Ω \equiv \partialΩ \cap \partial B(0,1) \ne \emptyset.$

It is well known that for

$α = 0,$

there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for

$α > 0$

and its asymptotic behavior as the parameter

$α$

approaches from below to a threshold

$α_0 ∈ (0,∞]$

for existence of a least energy solution.

Keywords: Least energy solutions, Hénon equation, limiting profile, Neumann boundary condition, weighted equations.

Mathematics Subject Classification: Primary: 35B33, 35B40, 35J15, 35J25, 35J61.

Citation: Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190

References:

[1]	Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi, Scu. Norm. Sup. Pisa, (1991), 9-25.
[2]	Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations, 20 (1995), 591-631. doi: 10.1080/03605309508821110.
[3]	Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and $L^∞$ -estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Differential Integral Equations, 8 (1995), 41-68.
[4]	Adimurthi, F. Pacella and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., 113 (1993), 318-350. doi: 10.1006/jfan.1993.1053.
[5]	M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Studies, 4 (2004), 453-467. doi: 10.1515/ans-2004-0406.
[6]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[7]	J. Byeon, S. Cho and J. Park, On the location of a peak point of a least energy solution for Hénon equation, Discrete Contin. Dyn. Syst., 30 (2011), 1055-1081. doi: 10.3934/dcds.2011.30.1055.
[8]	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.
[9]	J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states $\amalg$, J. Differential Equations, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.
[10]	J. Byeon and Z. Q. Wang, On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states, Journal of Functional Analysis, 274 (2018), 3325-3376. doi: 10.1016/j.jfa.2018.03.015.
[11]	D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Henon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.
[12]	D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Henon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.
[13]	J. Chabrowski and M. Willem, Least energy solutions of a critical Neumann problem with a weight, Calc. Var. Partial Differential Equations, 15 (2002), 421-431. doi: 10.1007/s00526-002-0101-0.
[14]	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.
[15]	D. G. Costa and P. M. Girão, Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbations, J. Differential Equations, 188 (2003), 164-202. doi: 10.1016/S0022-0396(02)00070-0.
[16]	M. Gazzini and E. Serra, The Neumann problem for the Henon 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.
[17]	D. Gilbarg and N. Trudinger, Elliptic Partial Differntial Equations of Second Order, 2^nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.
[18]	M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronomy and Astrophysics, 24 (1973), 229-238.
[19]	C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.
[20]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case Ⅱ, Rev. Mat. Iberoamericana, 1 (1985), 45-121. doi: 10.4171/RMI/12.
[21]	P. L. Lions, F. Pacella and M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math. J., 37 (1988), 301-324. doi: 10.1512/iumj.1988.37.37015.
[22]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.
[23]	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.
[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]	S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disc, Comm. Contemp. Math., 8 (2006), 823-839. doi: 10.1142/S0219199706002295.
[26]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326. doi: 10.1007/s00526-004-0302-9.
[27]	D. Smets, J. Su and M. Willem, Non-radial ground states for the Hénon eqaution, Communications in Contemporary Mathematics, 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[28]	D. Smets, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.
[29]	X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.
[30]	J. Wei and S. Yan, Infinitely many nonradial solutions for the Hénon equation with critical growth, Rev. Mat. Iberoamericana, 29 (2013), 997-1020. doi: 10.4171/RMI/747.

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, Scu. Norm. Sup. Pisa, (1991), 9-25.
[2]	Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations, 20 (1995), 591-631. doi: 10.1080/03605309508821110.
[3]	Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and $L^∞$ -estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Differential Integral Equations, 8 (1995), 41-68.
[4]	Adimurthi, F. Pacella and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., 113 (1993), 318-350. doi: 10.1006/jfan.1993.1053.
[5]	M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Studies, 4 (2004), 453-467. doi: 10.1515/ans-2004-0406.
[6]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[7]	J. Byeon, S. Cho and J. Park, On the location of a peak point of a least energy solution for Hénon equation, Discrete Contin. Dyn. Syst., 30 (2011), 1055-1081. doi: 10.3934/dcds.2011.30.1055.
[8]	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.
[9]	J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states $\amalg$, J. Differential Equations, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.
[10]	J. Byeon and Z. Q. Wang, On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states, Journal of Functional Analysis, 274 (2018), 3325-3376. doi: 10.1016/j.jfa.2018.03.015.
[11]	D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Henon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.
[12]	D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Henon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.
[13]	J. Chabrowski and M. Willem, Least energy solutions of a critical Neumann problem with a weight, Calc. Var. Partial Differential Equations, 15 (2002), 421-431. doi: 10.1007/s00526-002-0101-0.
[14]	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.
[15]	D. G. Costa and P. M. Girão, Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbations, J. Differential Equations, 188 (2003), 164-202. doi: 10.1016/S0022-0396(02)00070-0.
[16]	M. Gazzini and E. Serra, The Neumann problem for the Henon 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.
[17]	D. Gilbarg and N. Trudinger, Elliptic Partial Differntial Equations of Second Order, 2^nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.
[18]	M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronomy and Astrophysics, 24 (1973), 229-238.
[19]	C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.
[20]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case Ⅱ, Rev. Mat. Iberoamericana, 1 (1985), 45-121. doi: 10.4171/RMI/12.
[21]	P. L. Lions, F. Pacella and M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math. J., 37 (1988), 301-324. doi: 10.1512/iumj.1988.37.37015.
[22]	W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.
[23]	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.
[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]	S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disc, Comm. Contemp. Math., 8 (2006), 823-839. doi: 10.1142/S0219199706002295.
[26]	E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326. doi: 10.1007/s00526-004-0302-9.
[27]	D. Smets, J. Su and M. Willem, Non-radial ground states for the Hénon eqaution, Communications in Contemporary Mathematics, 4 (2002), 467-480. doi: 10.1142/S0219199702000725.
[28]	D. Smets, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.
[29]	X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.
[30]	J. Wei and S. Yan, Infinitely many nonradial solutions for the Hénon equation with critical growth, Rev. Mat. Iberoamericana, 29 (2013), 997-1020. doi: 10.4171/RMI/747.

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