Symmetry breaking results for problems with exponential growth in the unit disk

We investigate some asymptotic properties of extrema u(alpha) to the two-dimensional variational problem sup integral(B) (e(gamma u2) - 1)vertical bar x vertical bar(alpha)dx e epsilon H1/0(B) vertical bar vertical bar u vertical bar vertical bar= 1 as alpha + infinity. Here B is the unit disk of R-2 and 0 < gamma <= 4 pi is a given parameter. We prove that in a certain range of gamma's, the maximizers are not radial for alpha large.