For every singular weight |x|^{-2β} and every α below the first Hardy eigenvalue, the supremum of ∫_B e^{4π(1−β)u²}|x|^{-2β} dx over the improved Hardy unit ball is finite and attained.
Struwe, Global compactness properties of semilinear elliptic equation with critical exponential growth, J
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Extremal functions for a singular Hardy-Moser-Trudinger inequality
For every singular weight |x|^{-2β} and every α below the first Hardy eigenvalue, the supremum of ∫_B e^{4π(1−β)u²}|x|^{-2β} dx over the improved Hardy unit ball is finite and attained.