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Ground state solutions of Hamiltonian elliptic systems in dimension two

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  12 February 2019

Djairo G. de Figueiredo ,
João Marcos do Ó  and
Jianjun Zhang
Djairo G. de Figueiredo
Affiliation:
IMECC, Universidade Estadual de Campinas, C.P. 6065, 13081–970, Campinas, SP, Brazil (djairo@ime.unicamp.br)
João Marcos do Ó
Affiliation:
Federal University of Paraíba, 58051–900, João Pessoa-PB, Brazil (jmbo@pq.cnpq.br)
Jianjun Zhang
Affiliation:
College Math. and Statistics, Chongqing Jiaotong University, Chongqing400074PR China (zhangjianjun09@tsinghua.org.cn)
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Abstract

The aim of this paper is to study Hamiltonian elliptic system of the form 0.1

$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$
where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2
$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$
We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

Keywords

Hamiltonian elliptic systemTrudinger–Moser inequalityground stategeneralized Nehari manifold

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35J50: Variational methods for elliptic systems 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1737 - 1768
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Copyright
Copyright © Royal Society of Edinburgh 2019

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