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The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain

We consider the Dirichlet problem Δu+λu+|u|2*−2u = 0 in Ω, u = 0 on ∂Ω where Ω is a bounded smooth domain in ℝN, N⩾4, and 2* = 2N/(N−2) is the critical Sobolev exponent. We show that if Ω is invariant under an orthogonal involution then, for λ>0 sufficiently small, there is an effect of the equivariant topology of Ω on the number of solutions which change sign exactly once.