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

João Marcos do Ó; Bruno Ribeiro; Bernhard Ruf

doi:10.3934/dcds.2020138

Article Contents

2021, Volume 41, Issue 1: 277-296. Doi: 10.3934/dcds.2020138

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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

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

Received: October 2019

Early access: February 2020

Published: January 2021

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

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Abstract

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.

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

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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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