Advances in Differential Equations

Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $\Bbb R^3$

Zongming Guo and Juncheng Wei

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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 $\mathbb R^2$.

Article information

Source
Adv. Differential Equations, Volume 13, Number 7-8 (2008), 753-780.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867335

Mathematical Reviews number (MathSciNet)
MR2479029

Zentralblatt MATH identifier
1203.35018

Subjects
Primary: 35J40: Boundary value problems for higher-order elliptic equations
Secondary: 35B45: A priori estimates 35J60: Nonlinear elliptic equations 47J15: Abstract bifurcation theory [See also 34C23, 37Gxx, 58E07, 58E09]

Citation

Guo, Zongming; Wei, Juncheng. Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $\Bbb R^3$. Adv. Differential Equations 13 (2008), no. 7-8, 753--780. https://projecteuclid.org/euclid.ade/1355867335


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