Context. PDS 70 is a young (5.4 Myr), nearby (~113 pc) star hosting a known transition disk with a large gap. Recent observations with SPHERE and NACO in the near-infrared (NIR) allowed us to detect a planetary mass companion, PDS 70 b, within the disk cavity. Moreover, observations in H_α with MagAO and MUSE revealed emission associated to PDS 70 b and to another new companion candidate, PDS 70 c, at a larger separation from the star. PDS 70 is the only multiple planetary system at its formation stage detected so far through direct imaging. Aims. Our aim is to confirm the discovery of the second planet PDS 70 c using SPHERE at VLT, to further characterize its physical properties, and search for additional point sources in this young planetary system. Methods. We re-analyzed archival SPHERE NIR observations and obtained new data in Y, J, H and K spectral bands for a total of four different epochs. The data were reduced using the data reduction and handling pipeline and the SPHERE data center. We then applied custom routines (e.g., ANDROMEDA and PACO) to subtract the starlight. Results. We re-detect both PDS 70 b and c and confirm that PDS 70 c is gravitationally bound to the star. We estimate this second planet to be less massive than 5 M_Jup and with a T_eff around 900 K. Also, it has a low gravity with logg between 3.0 and 3.5 dex. In addition, a third object has been identified at short separation (~0.12′′) from the star and gravitationally bound to the star. Its spectrum is however very blue, meaning that we are probably seeing stellar light reflected by dust and our analysis seems to demonstrate that it is a feature of the inner disk. We cannot however completely exclude the possibility that it is a planetary mass object enshrouded by a dust envelope. In this latter case, its mass should be of the order of a few tens of M_⊕. Moreover, we propose a possible structure for the planetary system based on our data, and find that this structure cannot be stable on a long timescale.