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

Published online by Cambridge University Press:  26 January 2019

Alireza Khatib
Affiliation:
Departamento de Matemática, Universidade Federal do Amazonas, Manaus, Amazonas, 69077-000, Brazil (alireza@ufam.edu.br)
Liliane A. Maia
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília-DF, 70910-900, Brazil (lilimaia@unb.br)
Corresponding

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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