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We obtain a nigh optimal estimate for the first eigenvalue of two naturalweighted problems associated to the bilaplacian (and of a continuous family offourth-order elliptic operators in dimension $2$) in degenerating annuli (thatare central objects in bubble tree analysis) in all dimension. The estimatedepends only on the conformal class of the annulus. We also show that indimension $2$ and dimension $4$, the first eigenfunction (of the first problem)is never radial provided that the conformal class of the annulus is largeenough. The other result is a weighted Poincar\'e-type inequality in annuli forthose fourth-order operators. Applications to Morse theory are given.

Weighted Eigenvalue Problems for Fourth-Order Operators in Degenerating Annuli