2006 International Conference in Honor of Jacqueline Fleckinger.
Electronic Journal of Differential Equations,
Conference 16 (2007), pp. 103-128.
Title: Large radial solutions of a polyharmonic equation
with superlinear growth
Authors: J. Ildefonso Diaz (Univ. Complutense de Madrid, Spain)
Monica Lazzo (Univ. di Bari, Italy)
Paul G. Schmidt (Auburn Univ., AL, USA)
Abstract:
This paper concerns the equation $\Delta\!^m u=|u|^p$,
where $m\in\mathbb{N}$, $p\in(1,\infty)$, and $\Delta$ denotes
the Laplace operator in $\mathbb{R}^N\!$, for some $N\in\mathbb{N}$.
Specifically, we are interested in the structure of the set
$\mathcal{L}$ of all large radial solutions
on the open unit ball $B$ in $\mathbb{R}^N$.
In the well-understood second-order case,
the set $\mathcal{L}$ consists of exactly two solutions if
the equation is subcritical, of exactly one solution if it
is critical or supercritical. In the fourth-order case, we show that
$\mathcal{L}$ is homeomorphic to the unit circle $S^1$ if the equation
is subcritical, to $S^1$ minus a single point if it is critical
or supercritical. For arbitrary $m\in\mathbb{N}$, the set
$\mathcal{L}$ is a full $(m-1)$-sphere whenever the equation is subcritical.
We conjecture, but have not been able to prove in general, that
$\mathcal{L}$ is a punctured $(m-1)$-sphere whenever the equation
is critical or supercritical. These results and the conjecture
are closely related to the existence and uniqueness (up to scaling)
of entire radial solutions. Understanding the geometric
and topological structure of the set $\mathcal{L}$ allows precise
statements about the existence and multiplicity of large radial solutions
with prescribed center values
$u(0),\Delta u(0),\dots,\Delta\!^{m-1}u(0)$.
Published May 15, 2007.
Math Subject Classifications: 35J40, 35J60.
Key Words: Polyharmonic equation; radial solutions; entire solutions;
large solutions; existence and multiplicity; boundary blow-up.