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A CLASS OF CRITICAL KIRCHHOFF PROBLEM ON THE HYPERBOLIC SPACE $\mathbb{H}^{{\it n}}$

Part of: Partial differential equations Miscellaneous topics - Partial differential equations Partial differential equations on manifolds; differential operators Elliptic equations and systems

Published online by Cambridge University Press:  21 January 2019

PAULO CESAR CARRIÃO ,
AUGUSTO CÉSAR DOS REIS COSTA  and
OLIMPIO HIROSHI MIYAGAKI
PAULO CESAR CARRIÃO
Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, Minas Gerais, Brazil e-mail: pauloceca@gmail.com
AUGUSTO CÉSAR DOS REIS COSTA
Affiliation:
Faculdade de Matemática, Instituto de Ciências Exatas e Naturais, Universidade Federal do Pará, 66075-110 Belém, PA, Brazil e-mail: aug@ufpa.br
OLIMPIO HIROSHI MIYAGAKI*
Affiliation:
Departamento de Matemática, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, Minas Gerais, Brazil e-mail: ohmiyagaki@gmail.com
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Abstract

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We investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball $B_1(0)\subset \mathbb{R}^n$ Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.

MSC classification

Primary: 58J05: Elliptic equations on manifolds, general theory
Secondary: 35R01: Partial differential equations on manifolds 35J60: Nonlinear elliptic equations 35B33: Critical exponents

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 109 - 122
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019 

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