Quantitative Stability and Perturbation Effects in Conformal PDEs and Inequalities

Abstract: In this talk, we discuss our contributions to the quantitative stability of conformally invariant inequalities and to perturbation effects in semilinear P.D.E.s. We begin with the question of quantitative stability, first posed by Brezis–Lieb (1985) for the Sobolev inequality, and extend it to a Moser–Onofri type inequality recently proved by Chang–Gui (2023). In the spirit of Tobias König(2025), we further investigate the existence of Bianchi–Egnell type minimizers for the quantitative stability of the Hardy–Sobolev inequality. Finally, inspired by the seminal work of Brezis–Nirenberg (1983), we observe a sharp threshold separating the existence and nonexistence regimes for positive solutions to the critical exponent problem with logarithmic perturbations in hyperbolic space.