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EXISTENCE OF POSITIVE SOLUTION FOR INDEFINITE KIRCHHOFF EQUATION IN EXTERIOR DOMAINS WITH SUBCRITICAL OR CRITICAL GROWTH

Part of: Functional-differential and differential-difference equations Elliptic equations and systems Boundary value problems Qualitative theory Qualitative behavior

Published online by Cambridge University Press:  23 December 2016

G. M. FIGUEIREDO  and
D. C. DE MORAIS FILHO
G. M. FIGUEIREDO
Affiliation:
Universidade Federal do Pará, Faculdade de Matemática, CEP: 66075-110, Belém - Pa, Brazil email giovany@ufpa.br
D. C. DE MORAIS FILHO*
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58109-970, Campina Grande - Pb, Brazil email daniel@dme.ufcg.edu.br
*
✉daniel@dme.ufcg.edu.br
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Abstract

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Using variational methods and depending on a parameter $\unicode[STIX]{x1D706}$ we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$:

$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$
for the subcritical case ($\unicode[STIX]{x1D6FE}=0$) and also for the critical case ($\unicode[STIX]{x1D6FE}=1$).

Keywords

Variational methodsnonlocal problemsKirchhoff equationexterior domain

MSC classification

Primary: 45M20: Positive solutions
Secondary: 35J25: Boundary value problems for second-order elliptic equations 34B18: Positive solutions of nonlinear boundary value problems 34C11: Growth, boundedness 34K12: Growth, boundedness, comparison of solutions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 3 , December 2017 , pp. 329 - 340
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was partially supported by PROCAD/CASADINHO: 552101/2011-7 and CNPq/PQ 301242/2011-9; the second author was partially supported by PROCAD/Casadinho: 552.464/2011-2 and FNDE-PET/BRAZIL.

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