Electron. J. Diff. Eqns., Vol. 2003(2003), No. 37, pp. 1-13.

Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry

P. J. McKenna & Wolfgang Reichel

Positive entire solutions of the singular biharmonic equation $\Delta^2 u + u^{-q}=0$ in $\mathbb{R}^n$ with $q geater than 1$ 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.

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Joe McKenna
Department of Mathematics
University of Connecticut
Storrs, CT 06269, USA
email: mckenna@math.uconn.edu
Wolfgang Reichel
Mathematisches Institut
Universitat Basel
Rheinsprung 21, CH-4051 Basel, Switzerland
email: reichel@math.unibas.ch

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