Published Paper

Inserted: 16 feb 2006
Last Updated: 18 jan 2007

Year: 2006

Abstract:

We investigate entire radial solutions of the semilinear biharmonic equation $\Delta^2 u = \lambda \exp (u)$ in $\mathbb{R}^n$, $n\ge 5$, $\lambda >0$ being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of $\mathbb{R}^n$. In particular, they cannot be expanded as power series in the natural variable $s =\log
x
$. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to $-\infty$ as $
x
\to \infty$ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities \cite{GazzolaGrunau}, the entire singular solution $x\mapsto -4\log
x
$ plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case $n=5$.

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