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This paper is devoted to the study of the nonlinear elliptic problem with supercritical critical exponent (P ε ) : −Δu = K|u| 4/(n−2)+ε u in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ n , n ≥ 3, K is a C 3  positive function and ε is a positive real parameter. We show first that in dimension 3, for ε small, (P ε ) has no sign-changing solutions with low energy which blow up at two points. For n ≥ 4, we prove that there are no sign-changing solutions which blow up at two nearby points. We also show that (P ε ) has no bubble-tower sign-changing solutions.

Nonexistence Results of Sign-changing solutions for a Supercritical Problem of the Scalar Curvature Type