Non-radial Maximizers For Functionals With Exponential Non-linearity in ℝ2

Abstract

Abstract We consider the functional F : H1 0(B(0,1)) → ℝ where α > 0,p > 0, 1 < γ ≤ 2, and B(0,1) is the unit ball in ℝ2. We prove that for any p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2 no maximizer of F(u) on the unit ball in H1 0 is radially symmetric provided that cu 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 Hénon equation with Dirichlet boundary conditions.