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doi: 10.3934/dcds.2020138

Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions

João Marcos do Ó 1, , Bruno Ribeiro 1, and  Bernhard Ruf 2,,

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

* Corresponding author: bernhard.ruf@unimi.it

Received  October 2019 Published  February 2020

Fund Project: The three authors were partially supported by CNPq-Brazil Grants PVE 407099/2013-1

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 $
 is a bounded domain and
$ g $
 is a nonlinearity with exponential growth condition. We derive the maximal growth conditions allowed for
$ f $
, proving that it can be of exponential type, double-exponential type, or completely arbitrary, depending on the conditions required for
$ g $
. Under the hypothesis of arbitrary growth conditions or else when
$ f $
 has a double exponential growth, we prove existence of nontrivial solutions for the system.
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 & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020138
References:
[1]

R. Adams, Sobolev Spaces, in Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Krasnosel'skii and Y. Rutickii, Convex functions and Orlicz Spaces, P. Noordhoff, Ltd. Groningen, Netherlands, 1961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces, in Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Krasnosel'skii and Y. Rutickii, Convex functions and Orlicz Spaces, P. Noordhoff, Ltd. Groningen, Netherlands, 1961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Polynomial critical hyperbola and related growth conditions
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Figure 2.  The critical hyperbola in the exponential case
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