  Although M dwarfs are known for high levels of stellar activity, they are
ideal targets for the search of low-mass exoplanets with the radial velocity
(RV) method. We report the discovery of a planetary-mass companion around LSPM
J2116+0234 (M3.0 V) and confirm the existence of a planet orbiting GJ 686
(BD+18 3421; M1.0 V). The discovery of the planet around LSPM J2116+0234 is
based on CARMENES RV observations in the visual and near-infrared channels. We
confirm the planet orbiting around GJ 686 by analyzing the RV data spanning
over two decades of observations from CARMENES VIS, HARPS-N, HARPS, and HIRES.
We find planetary signals at 14.44 and 15.53 d in the RV data for LSPM
J2116+0234 and GJ 686, respectively. Additionally, the RV, photometric time
series, and various spectroscopic indicators show hints of variations of 42 d
for LSPM J2116+0234 and 37 d for GJ 686, which we attribute to the stellar
rotation periods. The orbital parameters of the planets are modeled with
Keplerian fits together with correlated noise from the stellar activity. A
mini-Neptune with a minimum mass of 11.8 Me orbits LSPM J2116+0234 producing an
RV semi-amplitude of 6.19 m/s, while a super-Earth of mass 6.6 Me orbits GJ 686
and produces an RV semi-amplitude of 3.0 m/s. Both LSPM J2116+0234 and GJ 686
have planetary companions populating the regime of exoplanets with masses lower
than 15 Me and orbital periods <20 d.
