Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 37, pp. 1-13.
Title: Radial solutions of singular nonlinear biharmonic equations and
applications to conformal geometry
Authors: P. J. McKenna (Univ. Connecticut, Storrs, USA)
Wolfgang Reichel (Univ. Basel, Switzerland)
Abstract:
Positive entire solutions of the singular biharmonic equation
$\Delta^2 u + u^{-q}=0$ in $\mathbb{R}^n$ with $q>1$ and $n\geq 3$
are considered. We prove that there are infinitely many radial
entire solutions with different growth rates close to quadratic.
If $u(0)$ is kept fixed we show that a unique minimal entire
solution exists, which separates the entire solutions from those
with compact support. For the special case $n=3$ and $q=7$
the function $U(r) = \sqrt{1/\sqrt{15}+r^2}$ is the minimal entire
solution if $u(0)=15^{-1/4}$ is kept fixed.
Submitted February 12, 2003. Published April 10, 2003.
Math Subject Classifications: 35J60.
Key Words: Singular biharmonic equation; conformal invariance.