Non-Symmetric low-index solutions for a symmetric boundary value problem

Code:
36/2010
Title:
Non-Symmetric low-index solutions for a symmetric boundary value problem
Date:
Thursday 11th November 2010
Author(s):
Arioli, G.; Koch, H.
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Abstract:
We consider the equation -Laplacian(u)=w*u^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.