  A search for non-resonant Higgs boson pair ($HH$) production is presented, in
which one of the Higgs bosons decays to a b-quark pair ($b\bar b$) and the
other decays to $WW^*$, $ZZ^*$, or $\tau^+\tau^-$, with in each case a final
state with $\ell^+\ell^- +$ neutrinos ($\ell = e, \mu$). The analysis targets
separately the gluon-gluon fusion and vector boson fusion production modes.
Data recorded by the ATLAS detector in proton-proton collisions at a
centre-of-mass energy of 13 TeV at the Large Hadron Collider, corresponding to
an integrated luminosity of $140\mathrm{fb}^{-1}$, are used in this analysis.
Events are selected to have exactly two $b$-tagged jets and two leptons with
opposite electric charge and missing transverse momentum in the final state.
These events are classified using multivariate analysis algorithms to separate
the $HH$ events from other Standard Model processes. No evidence of the signal
is found. The observed (expected) upper limit on the cross-section for
non-resonant Higgs boson pair production is determined to be 9.7 (16.2) times
the Standard Model prediction at 95% confidence level. The Higgs boson
self-interaction coupling parameter $\kappa_\lambda$ and the quadrilinear
coupling parameter $\kappa_{2V}$ are each separately constrained by this
analysis to be within the ranges ${[-6.2, 13.3]}$ and ${[-0.17, 2.4]}$,
respectively, at 95% confidence level, when all other parameters are fixed.
