Non-radial maximizers for functionals with exponential non-linearity in R-2

We consider the functional F:H-0(1)(B(0,1))-> R F(u)=integral(B(0,1)) vertical bar x vertical bar(alpha)(e(p vertical bar u vertical bar gamma)-1-p vertical bar u vertical bar(gamma))dx where alpha>0, p>0, 1<= 2, and B(0,1) is the unit ball in R-2. We prove that for any p>0, 1<2 and 0<4 pi, gamma=2 no maximizer of F(u) on the unit ball in H-0(1) is radially symmetric provided that alpha is large enough. This extends a result of Smets, Su and Willem concerning the existence of non-radial ground state solutions for the Rayleigh quotient related to the Henon equation with Dirichlet boundary conditions.