05-33
Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity
by Arioli,G.; Gazzola, F.; Grunau, H.-Ch.
Preprint series: 05-33, Preprints
The paper is published: J. Differ. Equations, 230 , 743 - 770 (2006).
- MSC:
- 35J60 Nonlinear PDE of elliptic type
- 35J30 General theory of higher-order, elliptic equations , See also {31A30, 31B30}
- 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
- 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
Notes: Copyright: Elsevier
The author(s) agree, that this abstract may be stored asfull text and distributed as such by abstracting services.