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2005-33

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

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).