  Massless minimally coupled quantum scalar field with an asymmetric self
interaction, $V(\phi)=\lambda \phi^4/4!+ \beta \phi^3/3! $ (with $\lambda >0$)
is considered in the $(3+1)$-dimensional inflationary de Sitter spacetime. This
potential is bounded from below irrespective of the sign of $\beta$. Earlier
computations mostly considered the quartic part. Our chief motivation behind
this study is to assess the vacuum expectation values of $V(\phi)$ and $\phi$,
both of which can be negative, and hence may lead to some screening of the
inflationary cosmological constant value. First using the in-in formalism, the
renormalised quantum correction to the cubic potential appearing in the
energy-momentum tensor is computed at two loop, which is the leading order in
this case. The quantum correction to the kinetic term at two loop are
subleading compared to the above result at late cosmological times. Next, using
some of these results we compute the renormalised vacuum expectation value of $
\phi$, by computing the tadpoles at ${\cal O}(\beta)$ and ${\cal O}(\lambda
\beta)$. Due to the appearance of the de Sitter isometry breaking logarithms,
the tadpoles cannot be completely renormalised away in this case, unlike the
flat spacetime. All these results, as expected, show secularly growing
logarithms at late cosmological times. We next use a recently proposed
renormalisation group inspired formalism to resum perturbative secular effects,
to compute a non-perturbative $\langle \phi \rangle$ at late cosmological
times. $\langle \phi \rangle$ turns out to be approximately one order of
magnitude less compared to the position of the classical minima
$\phi=-3\beta/\lambda$ of $V(\phi)$. Estimation on the possible screening of
the inflationary cosmological constant due to this $\langle \phi \rangle$ is
also presented.
