  It is well-known that stars with giant planets are on average more metal-rich
than stars without giant planets, whereas stars with detected low-mass planets
do not need to be metal-rich. With the aim of studying the weak boundary that
separates giant planets and brown dwarfs (BDs) and their formation mechanism,
we analyze the spectra of a sample of stars with already confirmed BD
companions both by radial velocity and astrometry. We employ standard and
automatic tools to perform an EW-based analysis and to derive chemical
abundances from CORALIE spectra of stars with BD companions. We compare these
abundances with those of stars without detected planets and with low-mass and
giant-mass planets. We find that stars with BDs do not have metallicities and
chemical abundances similar to those of giant-planet hosts but they resemble
the composition of stars with low-mass planets. The distribution of mean
abundances of $\alpha$-elements and iron peak elements of stars with BDs
exhibit a peak at about solar abundance whereas for stars with low-mass and
high-mass planets the [X$_\alpha$/H] and [X$_{\rm Fe}$/H] peak abundances
remain at $\sim -0.1$~dex and $\sim +0.15$~dex, respectively. We display these
element abundances for stars with low-mass and high-mass planets, and BDs
versus the minimum mass, $m_C \sin i$, of the most-massive substellar companion
in each system, and we find a maximum in $\alpha$-element as well as Fe-peak
abundances at $m_C \sin i \sim 1.35\pm 0.20$ jupiter masses. We discuss the
implication of these results in the context of the formation scenario of BDs in
comparison with that of giant planets.
