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The Hénon equation with a critical exponent under the Neumann boundary condition
Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea |
$ -Δ u + u = |x|^{α}u^{p}, \; u > 0 \;\text{in} \; Ω,\ \ \frac{\partial u}{\partial ν} = 0 \; \text{ on }\;\partial Ω,$ |
$Ω \subset B(0,1) \subset \mathbb{R}^n, n ≥ 3$, $α≥ 0$ and $\partial^*Ω \equiv \partialΩ \cap \partial B(0,1) \ne \emptyset.$ |
$α = 0,$ |
$α > 0$ |
$α$ |
$α_0 ∈ (0,∞]$ |
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, 2nd 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, 2nd 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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