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Let B be the unit disc in R^2, H be the completion of C_0^∞(B) under the norm ||u||_H =(_B | abla u|^2dx-_B u^2(1-|x|^2)^2dx)^12 ,uC_0^∞(B). Using blow-up analysis, we prove that for any γ4π, the supremum align*_uH,||u||_1,h1_B e^γ u^2dxalign* can be attained by some function u_0H with ||u_0||_1,h=1, where h is a decreasingly nonnegative, radially symmetric function, and satisfies a coercive condition. Namely there exists a constant δ>0 satisfying ||u||_1,h^2=\|u\|_H ^2-_B hu^2dx≥ δ\|u\|_H ^2, uH. This extends earlier results of Wang-Ye [1] and Yang-Zhu [2]. , number=4, journal=Journal of Partial Differential Equations, year=2020, month=May, pages=353–373