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BIFURCATION PROPERTIES FOR A CLASS OF CHOQUARD EQUATION IN WHOLE ℝ3

Part of: Difference and functional equations Difference equations Partial differential equations

Published online by Cambridge University Press:  11 July 2019

CLAUDIANOR O. ALVES ,
ROMILDO N. DE LIMA  and
ALÂNNIO B. NÓBREGA
CLAUDIANOR O. ALVES
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP: 58429-900, Brazil e-mails: coalves@mat.ufcg.edu.br, romildo@mat.ufcg.edu.br, alannio@mat.ufcg.edu.br
ROMILDO N. DE LIMA
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP: 58429-900, Brazil e-mails: coalves@mat.ufcg.edu.br, romildo@mat.ufcg.edu.br, alannio@mat.ufcg.edu.br
ALÂNNIO B. NÓBREGA
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP: 58429-900, Brazil e-mails: coalves@mat.ufcg.edu.br, romildo@mat.ufcg.edu.br, alannio@mat.ufcg.edu.br
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Abstract

This paper concerns the study of some bifurcation properties for the following class of Choquard-type equations: (P)

$$\left\{ {\begin{array}{*{20}{l}} { - \Delta u = \lambda f(x)\left[ {u + \left( {{I_\alpha }*f( \cdot )H(u)} \right)h(u)} \right],{\rm{ in }} \ {{\mathbb{R}}^3},}\\ {{{\lim }_{|x| \to \infty }}u(x) = 0,\quad u(x) > 0,\quad x \in {{\mathbb{R}}^3},\quad u \in {D^{1,2}}({{\mathbb{R}}^3}),} \end{array}} \right.$$
where ${I_\alpha }(x) = 1/|x{|^\alpha },\,\alpha \in (0,3),\,\lambda > 0,\,f:{{\mathbb{R}}^3} \to {\mathbb{R}}$ is a positive continuous function and h : ${\mathbb{R}} \to {\mathbb{R}}$ is a bounded Hölder continuous function. The main tools used are Leray–Schauder degree theory and a global bifurcation result due to Rabinowitz.

MSC classification

Primary: 39A28: Bifurcation theory
Secondary: 35A16: Topological and monotonicity methods
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020 , pp. 531 - 543
Copyright
© Glasgow Mathematical Journal Trust 2019

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