by Arioli,G., Gazzola, F., Grunau, H.-Ch..

**Series:** 2005-33, Preprints

- MSC:
- 35J60 Nonlinear elliptic equations
- 35J30 Higher-order elliptic equations
- 35J65 Nonlinear boundary value problems for linear elliptic equations
- 35J40 Boundary value problems for higher-order elliptic equations

**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. 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 $n=5$.

**Keywords:**

semilinear, biharmonic, supercritical, exponential growth, entire solutions, separatrix

**This paper was published in:**

J. Differ. Equations, 230 , 743 - 770 (2006).