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2008 Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $\Bbb R^3$
Zongming Guo, Juncheng Wei
Adv. Differential Equations 13(7-8): 753-780 (2008).

Abstract

We study the structure of solutions of the boundary-value problem \begin{equation} \tag*{(0.1)} \Delta^2 u=\frac{\lambda}{(1-u)^2} \;\; \mbox{in $B$}, \;\;\; u=\Delta u=0 \;\; \mbox{on $\partial B$} , \end{equation} where $\Delta^2$ is the biharmonic operator and $B \subset \mathbb R^3$ is the unit ball. We show that there are infinitely many turning points of the branch of the radial solutions of (0.1). The structure of solutions depends on the classification of the radial solutions of the equation \begin{equation} \tag*{(0.2)} -\Delta^2 u=u^{-2} \;\;\; \mbox{in $\mathbb R^3$}. \;\; \end{equation} This is in sharp contrast with the corresponding result in $\mathbb R^2$.

Citation

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Zongming Guo. Juncheng Wei. "Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $\Bbb R^3$." Adv. Differential Equations 13 (7-8) 753 - 780, 2008.

Information

Published: 2008
First available in Project Euclid: 18 December 2012

zbMATH: 1203.35018
MathSciNet: MR2479029

Subjects:
Primary: 35J40
Secondary: 35B45, 35J60, 47J15

Rights: Copyright © 2008 Khayyam Publishing, Inc.

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Vol.13 • No. 7-8 • 2008
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