## Advances in Differential Equations

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

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

#### 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