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Positive solutions of Kirchhoff-type non-local elliptic equation: a bifurcation approach

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  28 June 2017

Zhanping Liang ,
Fuyi Li  and
Junping Shi
Zhanping Liang
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan, 030006 Shanxi, People's Republic of China
Fuyi Li
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan, 030006 Shanxi, People's Republic of China
Junping Shi
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795, USA (jxshix@wm.edu)
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Extract

Positive solutions of a Kirchhoff-type nonlinear elliptic equation with a non-local integral term on a bounded domain in ℝN, N ⩾ 1, are studied by using bifurcation theory. The parameter regions of existence, non-existence and uniqueness of positive solutions are characterized by the eigenvalues of a linear eigenvalue problem and a nonlinear eigenvalue problem. Local and global bifurcation diagrams of positive solutions for various parameter regions are obtained.

Keywords

Kirchhoff-type equationpositive solutionbifurcation theory

MSC classification

Secondary: 35J60: Nonlinear elliptic equations 35B32: Bifurcation 35J25: Boundary value problems for second-order elliptic equations 35J62: Quasilinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 4 , August 2017 , pp. 875 - 894
Copyright
Copyright © Royal Society of Edinburgh 2017 

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