The standard model (SM) production of four top quarks ( tt¯tt¯ ) in proton-proton collisions is studied by the CMS Collaboration. The data sample, collected during the 2016-2018 data taking of the LHC, corresponds to an integrated luminosity of 137 fb-1 at a center-of-mass energy of 13 TeV . The events are required to contain two same-sign charged leptons (electrons or muons) or at least three leptons, and jets. The observed and expected significances for the tt¯tt¯ signal are respectively 2.6 and 2.7 standard deviations, and the tt¯tt¯ cross section is measured to be 12.6-5.2+5.8fb . The results are used to constrain the Yukawa coupling of the top quark to the Higgs boson, yt , yielding a limit of |yt/ytSM|<1.7 at 95% confidence level, where ytSM is the SM value of yt . They are also used to constrain the oblique parameter of the Higgs boson in an effective field theory framework, H^<0.12 . Limits are set on the production of a heavy scalar or pseudoscalar boson in Type-II two-Higgs-doublet and simplified dark matter models, with exclusion limits reaching 350-470 GeV and 350-550 GeV for scalar and pseudoscalar bosons, respectively. Upper bounds are also set on couplings of the top quark to new light particles.