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.
Citation: |
[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.![]() ![]() ![]() |