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Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach
Nonradial least energy solutions of the p-Laplace elliptic equations
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan |
We study the p-Laplace elliptic equations in the unit ball under the Dirichlet boundary condition. We call u a least energy solution if it is a minimizer of the Lagrangian functional on the Nehari manifold. A least energy solution becomes a positive solution. Assume that the nonlinear term is radial and it vanishes in $|x| <a$ and it is positive in $a<|x|<1$. We prove that if a is close enough to 1, then no least energy solution is radial. Therefore there exist both a positive radial solution and a positive nonradial solution.
References:
[1] |
M. Badiale and E. Serra,
Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.
doi: 10.1515/ans-2004-0406. |
[2] |
V. Barutello, S. Secchi and E. Serra,
A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728.
doi: 10.1016/j.jmaa.2007.10.052. |
[3] |
J. Byeon and Z.-Q. Wang,
On the Hénon equation: asymptotic profile of ground states, Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828.
doi: 10.1016/j.anihpc.2006.04.001. |
[4] |
J. Byeon and Z.-Q. Wang,
On the Hénon equation: Asymptotic profile of ground states, Ⅱ, J. Differential Equations, 216 (2005), 78-108.
doi: 10.1016/j.jde.2005.02.018. |
[5] |
M. Calanchi, S. Secchi and E. Terraneo,
Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525.
doi: 10.1016/j.jde.2008.06.018. |
[6] |
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. |
[7] |
J.-L. Chern and C.-S. Lin,
The symmetry of least-energy solutions for semilinear elliptic equations, J. Differential Equations, 187 (2003), 240-268.
doi: 10.1016/S0022-0396(02)00080-3. |
[8] |
K. Deimling,
Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
[9] |
P. Drábek and J. Milota,
Methods of Nonlinear Analysis: Applications to Differential Equations Second edition, Birkhäuser, Berlin, 2013. |
[10] |
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. |
[11] |
N. Hirano,
Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations, 247 (2009), 1311-1333.
doi: 10.1016/j.jde.2009.06.008. |
[12] |
R. Kajikiya,
Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), 1353-1362.
doi: 10.1090/S0002-9939-2011-11172-9. |
[13] |
R. Kajikiya,
Non-radial least energy solutions of the generalized Hénon equation, J. Differential Equations, 252 (2012), 1987-2003.
doi: 10.1016/j.jde.2011.08.032. |
[14] |
R. Kajikiya,
Nonradial positive solutions of the p-Laplace Emden-Fowler equation with sign-changing weight, Mathematische Nachrichten, 289 (2016), 290-299.
doi: 10.1002/mana.201500103. |
[15] |
R. Kajikiya, Symmetric and asymmetric solutions of p-Laplace elliptic equations in hollow domains, To appear in Adv. Nonlinear Stud. |
[16] |
R. A. Moore and Z. Nehari,
Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 (1959), 30-52.
doi: 10.1090/S0002-9947-1959-0111897-8. |
[17] |
R. S. Palais,
The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[18] |
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. |
[19] |
P. Pucci and J. Serrin,
The Maximum Principle Birkhäuser, Berlin, 2007. |
[20] |
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. |
[21] |
D. Smets, M. Willem and J. Su,
Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480.
doi: 10.1142/S0219199702000725. |
[22] |
P. Tolksdorf,
On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8 (1983), 773-817.
doi: 10.1080/03605308308820285. |
[23] |
J. L. Vazquez,
A strong maximum principle for some quasilinear elliptic equations, Appl Math Optim, 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[24] |
E. Zeidler,
Applied Functional Analysis: Main Principles and Their Applications Springer, New York, 1995. |
show all references
References:
[1] |
M. Badiale and E. Serra,
Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.
doi: 10.1515/ans-2004-0406. |
[2] |
V. Barutello, S. Secchi and E. Serra,
A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728.
doi: 10.1016/j.jmaa.2007.10.052. |
[3] |
J. Byeon and Z.-Q. Wang,
On the Hénon equation: asymptotic profile of ground states, Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828.
doi: 10.1016/j.anihpc.2006.04.001. |
[4] |
J. Byeon and Z.-Q. Wang,
On the Hénon equation: Asymptotic profile of ground states, Ⅱ, J. Differential Equations, 216 (2005), 78-108.
doi: 10.1016/j.jde.2005.02.018. |
[5] |
M. Calanchi, S. Secchi and E. Terraneo,
Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525.
doi: 10.1016/j.jde.2008.06.018. |
[6] |
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. |
[7] |
J.-L. Chern and C.-S. Lin,
The symmetry of least-energy solutions for semilinear elliptic equations, J. Differential Equations, 187 (2003), 240-268.
doi: 10.1016/S0022-0396(02)00080-3. |
[8] |
K. Deimling,
Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
[9] |
P. Drábek and J. Milota,
Methods of Nonlinear Analysis: Applications to Differential Equations Second edition, Birkhäuser, Berlin, 2013. |
[10] |
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. |
[11] |
N. Hirano,
Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations, 247 (2009), 1311-1333.
doi: 10.1016/j.jde.2009.06.008. |
[12] |
R. Kajikiya,
Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), 1353-1362.
doi: 10.1090/S0002-9939-2011-11172-9. |
[13] |
R. Kajikiya,
Non-radial least energy solutions of the generalized Hénon equation, J. Differential Equations, 252 (2012), 1987-2003.
doi: 10.1016/j.jde.2011.08.032. |
[14] |
R. Kajikiya,
Nonradial positive solutions of the p-Laplace Emden-Fowler equation with sign-changing weight, Mathematische Nachrichten, 289 (2016), 290-299.
doi: 10.1002/mana.201500103. |
[15] |
R. Kajikiya, Symmetric and asymmetric solutions of p-Laplace elliptic equations in hollow domains, To appear in Adv. Nonlinear Stud. |
[16] |
R. A. Moore and Z. Nehari,
Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 (1959), 30-52.
doi: 10.1090/S0002-9947-1959-0111897-8. |
[17] |
R. S. Palais,
The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[18] |
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. |
[19] |
P. Pucci and J. Serrin,
The Maximum Principle Birkhäuser, Berlin, 2007. |
[20] |
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. |
[21] |
D. Smets, M. Willem and J. Su,
Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480.
doi: 10.1142/S0219199702000725. |
[22] |
P. Tolksdorf,
On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8 (1983), 773-817.
doi: 10.1080/03605308308820285. |
[23] |
J. L. Vazquez,
A strong maximum principle for some quasilinear elliptic equations, Appl Math Optim, 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[24] |
E. Zeidler,
Applied Functional Analysis: Main Principles and Their Applications Springer, New York, 1995. |
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