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A note on a positive solution of a null mass nonlinear field equation in exterior domains

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

Published online by Cambridge University Press:  26 January 2019

Alireza Khatib  and
Liliane A. Maia
Alireza Khatib
Affiliation:
Departamento de Matemática, Universidade Federal do Amazonas, Manaus, Amazonas, 69077-000, Brazil ([email protected])
Liliane A. Maia
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília-DF, 70910-900, Brazil ([email protected])
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Abstract

We consider the Null Mass nonlinear field equation (𝒫)

$$\left\{ {\matrix{ {-\Delta u = f(u){\rm in}\;\;\Omega } \hfill \hfill \hfill \hfill \cr {u > 0} \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \cr {u \vert_{\partial \Omega } = 0} \cr } } \right.$$
where ${\open R}^N \setminus \Omega $ is a bounded regular domain. The existence of a bound state solution is established in situations where this problem does not have a ground state.

Keywords

Nonlinear null mass equationexterior domainvariational methods

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35J20: Variational methods for second-order elliptic equations 35B09: Positive solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 841 - 870
Copyright
Copyright © Royal Society of Edinburgh 2019

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