International audienceCold cores are an early step of star formation, characterized by densities > 10$^4$ cm$^{-3}$, low temperatures (< 15 K), and very low external UV radiation. We investigate the physico-chemical processes at play to tracing the origin of molecules that are predominantly formed via reactions on dust grain surfaces. We observed the cold core LDN 429-C with the NOEMA interferometer and the IRAM 30m single dish telescope in order to obtain the gas-phase abundances of key species, including CO and CH$_3$OH. Comparing the observed gas phase of methanol to its solid phase previously observed with Spitzer allows us to put quantitative constraints on the efficiency of the non-thermal desorption of this species. With physical par...