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Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. (English) Zbl 1459.35144

Summary: In this paper we deal with the following class of Hamiltonian elliptic systems \[\left\{\begin{array}{lcl} -\Delta u=g(v)&\text{ in }\Omega,\\ -\Delta v=f(u)&\text{ in }\Omega,\\ u=v= 0&\text{ on }\partial\Omega, \end{array}\right. \] 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.

MSC:

35J57 Boundary value problems for second-order elliptic systems
35J50 Variational methods for elliptic systems
35B33 Critical exponents in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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