Information about the Higgs potential from the self-couplings of the detected Higgs boson is crucial for a better understanding of the mechanism of electroweak symmetry breaking and of the electroweak phase transition in the early Universe. The latter is of particular importance in the quest for explaining the observed asymmetry between matter and anti-matter in the Universe. It will be shown that although the present bounds on the trilinear Higgs self-coupling from Higgs pair production at the LHC are still relatively weak, they already provide a new way for probing so far unconstrained parameter regions of extended Higgs sectors. The parameter region of extended Higgs sectors giving rise to a strong first-order electroweak phase transition in the early Universe and potentially detectable gravitational wave signals is typically correlated with a certain mass splitting among the additional Higgs bosons, giving rise both to to a significant enhancement of the trilinear Higgs-boson self-coupling compared to the SM value and to a characteristic “smoking gun’’ signature in the search for additional Higgs boson. The prospects for exploiting the process of triple Higgs-boson production at the LHC for constraining both the trilinear and quartic Higgs boson self-couplings are briefly discussed. Join Zoom Meetinghttps://infn-it.zoom.us/j/96342145853?pwd=NzlkbUhPOTR4MmVFTXQ4dDhDK09lQT09Meeting ID: 963 4214 5853Passcode: 667657