  We study the nonlinear regime of large scale structure formation considering
a dynamical dark energy (DE) component determined by a Steep Equation of State
parametrization (SEoS) $w(z)=w_0+w_i\frac{(z/z_T)^q}{1+(z/z_T)^q}$. In order to
perform the model exploration at low computational cost, we modified the public
code L-PICOLA. We incorporate the DE model by means of the first and
second-order matter perturbations in the Lagrangian frame and the expansion
parameter. We analyze deviations of SEoS models with respect to $\Lambda$CDM in
the non-linear matter power spectrum ($P_k$), the halo mass function (HMF), and
the two-point correlation function (2PCF). On quantifying the nature of steep
(SEoS-I) and smooth transitions in DE field (CPL-lim), no signature of steep
transition is observed, rather found the overall impact of DE behaviors in
$P_k$ at level of $\sim 2-3\%$ and $\sim 3-4\%$ differences w.r.t $\Lambda$CDM
at $z=0$ respectively. HMF shows the possibility to distinguish between the
models at the high mass ends. The best-fitted model assuming only background
and linear perturbations dubbed as SEoS-II largely deviates from $\Lambda$CDM
and current observations on studying the nonlinear growth. This large deviation
in SEoS-II also quantified the combined effect of the dynamical DE and the
larger amount of matter contained, $\Omega_{m0}$ and $H_{0}$ accordingly. 2PCF
results are relatively robust with $\sim 1-2 \%$ deviation for SEoS-I and
CPL-lim and a significant deviation for SEoS-II throughout $r$ from
$\Lambda$CDM. Finally, we conclude that the search for viable DE models (like
the SEoS) must include non-linear growth constraints.
