Sixth Mississippi State Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 15 (2007), pp. 97-106.
Title: A pattern formation problem on the sphere
Authors: Clara E. Garza-Hume (Univ. Nacional Autonoma de Mexico)
Pablo Padilla (Univ. Nacional Autonoma de Mexico)
Abstract:
We consider a semi-linear elliptic equation on the sphere
$\mathbf{S}^n \subset \mathbb{R}^{n+1}$ with $n$ odd and
subcritical nonlinearity. We show that given any positive
integer $k$, if the exponent $p$ of the nonlinear term
is sufficiently close to the critical Sobolev exponent $p^*$,
then there exists a positive solution with $k$ peaks. Moreover,
the minimum energy solutions with $k$ peaks are such that the centers
of these concentrations converge as $p\to p^*$ to the solution of
an underlying geometrical problem, namely,
arranging $k$ points on $\mathbf{S}^n$ so they are as far away
from each other as possible.
Published February 28, 2007.
Math Subject Classifications: 35B33, 35J20.
Key Words: Semilinear elliptic equation; sphere packing;
critical Sobolev exponent; pattern formation.