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Polynomial critical hyperbola and related growth conditions
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January
2021, 41(1): 277-296.
doi: 10.3934/dcds.2020138
Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions
| 1. | Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa-PB, Brazil |
| 2. | Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy |
In this paper we deal with the following class of Hamiltonian elliptic systems
| $ \begin{equation*} \left\{\begin{array}{lcl} -\Delta u\ = g(v)&\mbox{in}&\Omega,\\ -\Delta v\ = f(u)&\mbox{in}&\Omega,\\ u\ = \ v = \ 0&\mbox{on}&\partial\Omega, \end{array}\right. \end{equation*} $ |
where
| $ \Omega\subset \mathbb{R}^2 $ |
| $ g $ |
| $ f $ |
| $ g $ |
| $ f $ |
Keywords:
Hamiltonian systems,
critical hyperbola,
elliptic systems,
exponential growth,
Trudinger-Moser inequality.
Mathematics Subject Classification:
Primary: 35A15, 35J50; Secondary: 35B33, 35B38.
Citation:
João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete and Continuous Dynamical Systems,
2021, 41
(1)
: 277-296.
doi: 10.3934/dcds.2020138
![]()
References:
| [1] |
R. Adams, Sobolev Spaces, in Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. |
| [2] |
H. Brézis and S. Wainger,
A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
| [3] |
D. Cassani and C. Tarsi,
Existence of solitary waves for supercritical Schrödinger systems in dimension two, Calc. Var. Partial Differential Equations, 54 (2015), 1673-1704.
doi: 10.1007/s00526-015-0840-3. |
| [4] |
A. Cianchi,
A Sharp Embedding Theorem for Orlicz-Sobolev Spaces, Indiana University Mathematics Journal, 45 (1996), 39-65.
doi: 10.1512/iumj.1996.45.1958. |
| [5] |
D. de Figueiredo and P. Felmer,
On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 99-116.
doi: 10.1090/S0002-9947-1994-1214781-2. |
| [6] |
D. de Figueiredo, J. M. do Ó and B. Ruf,
Critical and subcritical elliptic systems in dimension two, Indiana University Mathematics Journal, 53 (2004), 1037-1053.
doi: 10.1512/iumj.2004.53.2402. |
| [7] |
D. de Figueiredo, J. M. do Ó and B. Ruf,
An Orlicz-space approach to superlinear elliptic systems, J. Funct. Anal., 224 (2005), 471-496.
doi: 10.1016/j.jfa.2004.09.008. |
| [8] |
D. de Figueiredo, O. Miyagaki and B. Ruf,
Elliptic equations in $ \mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
| [9] |
D. de Figueiredo and B. Ruf,
Elliptic systems with nonlinearities of arbitrary growth, Mediterr. J. Math., 1 (2004), 417-431.
doi: 10.1007/s00009-004-0021-7. |
| [10] |
S. Hencl,
A sharp form of an embedding into exponential and doube exponential spaces, J. Funct. Anal.., 204 (2003), 196-227.
doi: 10.1016/S0022-1236(02)00172-6. |
| [11] |
J. Hulshof and R. van der Vorst,
Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.
doi: 10.1006/jfan.1993.1062. |
| [12] |
M. Krasnosel'skii and Y. Rutickii, Convex functions and Orlicz Spaces, P. Noordhoff, Ltd. Groningen, Netherlands, 1961. |
| [13] |
M. Rao and Z. Ren, Theory of Orlicz Spaces, in Monographs and Textbooks in Pure and Applied Mathematics, 146 Marcel Dekker, Inc., New York, 1991. |
| [14] |
B. Ruf, Lorentz-Sobolev spaces and nonlinear elliptic systems, in Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, Birkh user, Basel, 2006,471–489.
doi: 10.1007/3-7643-7401-2_32. |
show all references
References:
| [1] |
R. Adams, Sobolev Spaces, in Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. |
| [2] |
H. Brézis and S. Wainger,
A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
| [3] |
D. Cassani and C. Tarsi,
Existence of solitary waves for supercritical Schrödinger systems in dimension two, Calc. Var. Partial Differential Equations, 54 (2015), 1673-1704.
doi: 10.1007/s00526-015-0840-3. |
| [4] |
A. Cianchi,
A Sharp Embedding Theorem for Orlicz-Sobolev Spaces, Indiana University Mathematics Journal, 45 (1996), 39-65.
doi: 10.1512/iumj.1996.45.1958. |
| [5] |
D. de Figueiredo and P. Felmer,
On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 99-116.
doi: 10.1090/S0002-9947-1994-1214781-2. |
| [6] |
D. de Figueiredo, J. M. do Ó and B. Ruf,
Critical and subcritical elliptic systems in dimension two, Indiana University Mathematics Journal, 53 (2004), 1037-1053.
doi: 10.1512/iumj.2004.53.2402. |
| [7] |
D. de Figueiredo, J. M. do Ó and B. Ruf,
An Orlicz-space approach to superlinear elliptic systems, J. Funct. Anal., 224 (2005), 471-496.
doi: 10.1016/j.jfa.2004.09.008. |
| [8] |
D. de Figueiredo, O. Miyagaki and B. Ruf,
Elliptic equations in $ \mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
| [9] |
D. de Figueiredo and B. Ruf,
Elliptic systems with nonlinearities of arbitrary growth, Mediterr. J. Math., 1 (2004), 417-431.
doi: 10.1007/s00009-004-0021-7. |
| [10] |
S. Hencl,
A sharp form of an embedding into exponential and doube exponential spaces, J. Funct. Anal.., 204 (2003), 196-227.
doi: 10.1016/S0022-1236(02)00172-6. |
| [11] |
J. Hulshof and R. van der Vorst,
Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.
doi: 10.1006/jfan.1993.1062. |
| [12] |
M. Krasnosel'skii and Y. Rutickii, Convex functions and Orlicz Spaces, P. Noordhoff, Ltd. Groningen, Netherlands, 1961. |
| [13] |
M. Rao and Z. Ren, Theory of Orlicz Spaces, in Monographs and Textbooks in Pure and Applied Mathematics, 146 Marcel Dekker, Inc., New York, 1991. |
| [14] |
B. Ruf, Lorentz-Sobolev spaces and nonlinear elliptic systems, in Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, Birkh user, Basel, 2006,471–489.
doi: 10.1007/3-7643-7401-2_32. |
Figure 2.
The critical hyperbola in the exponential case
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