Sixth Mississippi State Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 15 (2007), pp. 97-106.

A pattern formation problem on the sphere

Clara E. Garza-Hume, Pablo Padilla

Abstract:
We consider a semi-linear elliptic equation on the sphere $\hbox{\bf S}^n \subset \mathbb{R}^{n+1}$ 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 $\hbox{\bf S}^n$ 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.

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Clara E. Garza-Hume
IIMAS-FENOMEC, Universidad Nacional Autónoma de México
Circuito Escolar, Cd. Universitaria 04510
México D. F., México
email: clara@mym.iimas.unam.mx
Pablo Padilla
IIMAS-FENOMEC, Universidad Nacional Autónoma de México
Circuito Escolar, Cd. Universitaria 04510
México D. F., México
email: pablo@mym.iimas.unam.mx

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