My question is that if a person is involved in sex with several girls, all of whom yield to his desires even though they are clearly being exploited, should we complain about this to any government authority because he is defiling the atmosphere of a place or should we hide all this?
You have described a very sad state of affairs. The crime this criminal seems to be obsessed with is indeed awfully disgusting and heinous.
As far as the action that you should take is considered, I would say that you are the best judge in this situation since you are in a position to better understand the circumstances. I would, however, like to bring to your notice some facts that one must consider before embarking on such an expedition.
Keeping in view the spirit of Islamic teachings and what the Prophet (sws) is reported to have said in this regard, the crime of those who stumble and commit adultery should be kept hidden from society in order to give them a chance to mend their ways. However, if a person is of the view that this will allow the couple to go astray further and lead a complete life of sin, then he may report it to the authorities. In this case he will have to produce four eyewitnesses in the court of law to prove the crime. These witnesses would testify that they have actually witnessed the crime committed. In case of failure to produce four eyewitnesses, the plaintiff shall stand punishable with eighty stripes under the law of Qadhf (24:4). Again, this provision of four witnesses has been so promulgated by the Almighty so that this crime is kept hidden from society and that the criminals be given a chance to mend their ways; and thus be saved from public humiliation.
However, as far as those people are concerned who habitually commit this crime, the society does become alive to their crime. In their case, the Qur’ān requires that only four witnesses be presented, who would testify to their ill ways just to comply with the proceedings of a court. After their crime stands proven formally, the Qur’ān demands that they be administered an exemplary punishment, which ranges from executing the criminal in an exemplary manner to exiling him to another area. (5:33)
However, I think that this second situation has not yet occurred in the case you have referred to. The society is not yet aware of the ill ways of the criminal. Thus, in case you choose to go to the court and register your report, you will surely have to produce four eyewitnesses as I have described above. It is only after the nature of this crime is exposed to the court that they declare him a habitual criminal and choose to administer any of the punishments as ordained in 5:33-4 for those who spread disorder in the society.