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In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in ℝ4{\mathbb{R}^{4}}. We also give a new Sobolev compact embedding which states W2,2⁢(ℝ4){W^{2,2}(\mathbb{R}^{4})} is compactly embedded into Lp⁢(ℝ4,|x|-β⁢d⁢x){L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)} for p≥2{p\geq 2} and 0<β<4{0<\beta<4}. As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity: