Abstract In this paper, we show that the minimizing problem Λ s , N , k , α = inf u ∈ H ˙ s ⁢ ( ℝ N ) , u ≢ 0 ⁡ ∫ ℝ N | ( - Δ ) s 2 ⁢ u ⁢ ( x ) | 2 ⁢ 𝑑 x ( ∫ ℝ N | u ⁢ ( x ) | 2 s , α * | y | α ⁢ 𝑑 x ) 2 2 s , α * $\Lambda_{s,N,k,\alpha}=\inf_{u\in\dot{H}^{s}(\mathbb{R}^{N}),u\not\equiv 0}% \frac{\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u(x)|^{2}\,dx}{\bigl{(}\int% _{\mathbb{R}^{N}}\frac{|u(x)|^{2^{*}_{s,\alpha}}}{|y|^{\alpha}}\,dx\bigr{)}^{% \frac{2}{2^{*}_{s,\alpha}}}}$ is achieved by a positive, cylindrically symmetric and strictly decreasing function u ⁢ ( x ) ${u(x)}$ provided 0 < s < N 2 ${0<s<\frac{N}{2}}$ , 0 < α < 2 ⁢ s ${0<\alpha<2s}$ , where x = ( y , z ) ∈ ℝ k × ℝ N - k ${x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}}$ and 2 s , α * = 2 ⁢ ( N - α ) N - 2 ⁢ s ${2^{*}_{s,\alpha}=\frac{2(N-\alpha)}{N-2s}}$ . Decaying laws for the minimizer u are also established.