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

  • * Corresponding author: bernhard.ruf@unimi.it

    * Corresponding author: bernhard.ruf@unimi.it
The three authors were partially supported by CNPq-Brazil Grants PVE 407099/2013-1
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  • 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.

    Mathematics Subject Classification: Primary: 35A15, 35J50; Secondary: 35B33, 35B38.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Polynomial critical hyperbola and related growth conditions

    Figure 2.  The critical hyperbola in the exponential case

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