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We construct blowing-up sign-changing solutions to some nonlinear criticalequations by glueing a standard bubble to a degenerate function. We develop amethod based on analyticity to perform the glueing when the critical manifoldof solutions is degenerate and no Bianchi--Egnell type condition holds.

Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold