Gianni Arioli, Hans Koch
Non-symmetric low-index solutions 
for a symmetric boundary value problem
(1899K, plain TeX, with eps figures)

ABSTRACT.  We consider the equation &Delta;<i>u=wu</i><sup>3</sup> 
on a square domain in R<sup>2</sup>, with Dirichlet boundary conditions, 
where <i>w</i> is a given positive function 
that is invariant under all (Euclidean) symmetries of the square. 
This equation is shown to have a solution <i>u</i>, 
with Morse index 2, that is neither symmetric nor antisymmetric 
with respect to any nontrivial symmetry of the square. 
Part of our proof is computer-assisted. 
An analogous result is proved for index 1.