Gianni Arioli, Hans Koch
Some symmetric boundary value problems and non-symmetric solutions
(408K, plain TeX, with eps figures)

ABSTRACT.  We consider the equation &Delta;<i>u=wf &prime;(u)</i> 
on a symmetric bounded domain in R<sup>n</sup> 
with Dirichlet boundary conditions. 
Here <i>w</i> is a positive function or measure 
that is invariant under the (Euclidean) symmetries of the domain. 
We focus on solutions <i>u</i> that are positive and/or 
have a low Morse index. 
Our results are concerned with the existence of non-symmetric 
solutions and the non-existence of symmetric solutions. 
In particular, we construct a solution <i>u</i> for the disk in R<sup>2</sup> 
that has index 2 and whose modulus <i>|u|</i> 
has only one reflection symmetry.