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Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential

We study the nonlinear Schrödinger equation −Δu + λa(x)u = μu + u2−1, u ∈ R , with critical exponent 2∗ = 2N/(N − 2), N ≥ 4, where a ≥ 0 has a potential well and is invariant under an orthogonal involution of R . Using variational methods we establish existence and multiplicity of solutions which change sign exactly once. These solutions localize near the potential well for μ small and λ large.