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Let D 1,2 (R N ) denote the closure of C ∞ 0 (R N ) in the norm ∥u∥ 2 D 1,2 (R N ) = |∇u| 2 . Let N > 3 and define the constants α N = N(N - 2) and CN = (N(N - 2)) N-2 4 . Let K ∈ C2(RN). We consider the following problem for e > 0: Find u ∈ D 1,2 (R N ) solving: (P e ) -Δu = α N (1+eK(x))u N+2 N-2 , in R N . We show an exact multiplicity result for (Pe) for all small e > 0.

Exact multiplicity result for the perturbed scalar curvature problem in $\mathbb{R}^N $ $(N \geq 3)$