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 Δ*u=wu*^{3}
on a square domain in R^{2}, with Dirichlet boundary conditions,
where *w* is a given positive function
that is invariant under all (Euclidean) symmetries of the square.
This equation is shown to have a solution *u*,
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.